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Properties

Label 4-508288-1.1-c1e2-0-4
Degree $4$
Conductor $508288$
Sign $-1$
Analytic cond. $32.4088$
Root an. cond. $2.38597$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 3·11-s − 8·13-s + 16-s + 11·17-s − 5·19-s − 3·22-s − 4·25-s + 8·26-s − 4·31-s − 32-s − 11·34-s − 37-s + 5·38-s − 12·43-s + 3·44-s + 10·49-s + 4·50-s − 8·52-s + 12·53-s + 4·62-s + 64-s + 11·68-s − 9·71-s + 13·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.904·11-s − 2.21·13-s + 1/4·16-s + 2.66·17-s − 1.14·19-s − 0.639·22-s − 4/5·25-s + 1.56·26-s − 0.718·31-s − 0.176·32-s − 1.88·34-s − 0.164·37-s + 0.811·38-s − 1.82·43-s + 0.452·44-s + 10/7·49-s + 0.565·50-s − 1.10·52-s + 1.64·53-s + 0.508·62-s + 1/8·64-s + 1.33·68-s − 1.06·71-s + 1.52·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 508288 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 508288 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(508288\)    =    \(2^{7} \cdot 11 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(32.4088\)
Root analytic conductor: \(2.38597\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 508288,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
19$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.17.al_cg
23$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.23.a_z
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.b_s
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.m_dt
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.47.a_k
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \) 2.59.a_bw
61$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.61.a_abl
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.j_ec
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.73.an_ha
79$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.p_ia
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.m_dy
89$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.89.a_ah
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.97.a_eg
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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

−8.190038446096234349329366492475, −7.87412899814022232815841220470, −7.29014413863468359330027317203, −7.17489714958834808563278964453, −6.61940574229480652017425138757, −5.94593937426796753768898862994, −5.46004485317344201667036216616, −5.20978336178999411113432460774, −4.35804599916004841764213471880, −3.88840585869286160042509916466, −3.24413323473804413844418683625, −2.61014540491342048494986988146, −1.95357836156256135390786536043, −1.20614804964089655967947503783, 0, 1.20614804964089655967947503783, 1.95357836156256135390786536043, 2.61014540491342048494986988146, 3.24413323473804413844418683625, 3.88840585869286160042509916466, 4.35804599916004841764213471880, 5.20978336178999411113432460774, 5.46004485317344201667036216616, 5.94593937426796753768898862994, 6.61940574229480652017425138757, 7.17489714958834808563278964453, 7.29014413863468359330027317203, 7.87412899814022232815841220470, 8.190038446096234349329366492475

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