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Properties

Label 4-238e2-1.1-c1e2-0-0
Degree $4$
Conductor $56644$
Sign $1$
Analytic cond. $3.61167$
Root an. cond. $1.37856$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 2·5-s − 3·6-s + 4·7-s − 8-s + 3·9-s + 2·10-s + 5·11-s − 6·13-s + 4·14-s − 6·15-s − 16-s + 17-s + 3·18-s + 2·19-s − 12·21-s + 5·22-s + 8·23-s + 3·24-s + 5·25-s − 6·26-s − 12·29-s − 6·30-s + 4·31-s − 15·33-s + 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 0.894·5-s − 1.22·6-s + 1.51·7-s − 0.353·8-s + 9-s + 0.632·10-s + 1.50·11-s − 1.66·13-s + 1.06·14-s − 1.54·15-s − 1/4·16-s + 0.242·17-s + 0.707·18-s + 0.458·19-s − 2.61·21-s + 1.06·22-s + 1.66·23-s + 0.612·24-s + 25-s − 1.17·26-s − 2.22·29-s − 1.09·30-s + 0.718·31-s − 2.61·33-s + 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56644\)    =    \(2^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(3.61167\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.446972218\)
\(L(\frac12)\) \(\approx\) \(1.446972218\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
17$C_2$ \( 1 - T + T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.d_g
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ap
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bp
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_cb
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_bh
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.71.k_gl
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.89.j_ai
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.97.ai_ic
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.33215273688516538139917971337, −11.96273322091326752014739687288, −11.31778185250371593533819143476, −11.10346912724920771704623403289, −10.93987544091937897929392191298, −10.02361498485272489352378580487, −9.593442456795073556139495875336, −9.004503754576572914638065383805, −8.754816157285502806737520237560, −7.46569015271377898963750040517, −7.40088361548313374083389975590, −6.71751317824741394404123785391, −5.92523756176559241037597442808, −5.61456093327685998667253858574, −5.29445375276360596244330334002, −4.58468497181050225052907947926, −4.37662829544461270266251944664, −3.14303983767728882414526998291, −2.08523693388849276973154916589, −1.08766844031086332498433508086, 1.08766844031086332498433508086, 2.08523693388849276973154916589, 3.14303983767728882414526998291, 4.37662829544461270266251944664, 4.58468497181050225052907947926, 5.29445375276360596244330334002, 5.61456093327685998667253858574, 5.92523756176559241037597442808, 6.71751317824741394404123785391, 7.40088361548313374083389975590, 7.46569015271377898963750040517, 8.754816157285502806737520237560, 9.004503754576572914638065383805, 9.593442456795073556139495875336, 10.02361498485272489352378580487, 10.93987544091937897929392191298, 11.10346912724920771704623403289, 11.31778185250371593533819143476, 11.96273322091326752014739687288, 12.33215273688516538139917971337

Graph of the $Z$-function along the critical line