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Properties

Label 4-2312e2-1.1-c1e2-0-13
Degree $4$
Conductor $5345344$
Sign $1$
Analytic cond. $340.823$
Root an. cond. $4.29667$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 2·7-s + 2·9-s − 2·11-s − 8·15-s − 4·19-s − 4·21-s − 2·23-s + 2·25-s + 6·27-s − 4·29-s + 2·31-s − 4·33-s + 8·35-s + 4·37-s − 4·41-s − 12·43-s − 8·45-s + 8·47-s − 6·49-s − 4·53-s + 8·55-s − 8·57-s + 20·59-s + 4·61-s − 4·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 0.755·7-s + 2/3·9-s − 0.603·11-s − 2.06·15-s − 0.917·19-s − 0.872·21-s − 0.417·23-s + 2/5·25-s + 1.15·27-s − 0.742·29-s + 0.359·31-s − 0.696·33-s + 1.35·35-s + 0.657·37-s − 0.624·41-s − 1.82·43-s − 1.19·45-s + 1.16·47-s − 6/7·49-s − 0.549·53-s + 1.07·55-s − 1.05·57-s + 2.60·59-s + 0.512·61-s − 0.503·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5345344\)    =    \(2^{6} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(340.823\)
Root analytic conductor: \(4.29667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5345344,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
17 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_s
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_w
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_cg
37$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_ac
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dy
47$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_be
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.59.au_hq
61$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_bu
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$D_{4}$ \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.71.o_he
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_dy
79$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_fi
83$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_ha
89$D_{4}$ \( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.89.y_lq
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.530331281265960957245710013040, −8.431635679106594336422474492883, −7.975669815462529350073322669559, −7.968654378424625052964344745597, −7.29304921297364302984044329819, −7.01440553160506986767525306197, −6.72304875595580026377835859686, −6.14932349097873625968388235514, −5.63225662549928117828236169746, −5.24971375726574680916279888378, −4.41998737654410291314220734517, −4.25775029536186691313102325812, −3.97545217050221942522207015573, −3.39983577575754223695834510551, −2.94471000703079211711062979119, −2.76966127457592119584350767723, −1.99796869686072108032830957604, −1.33929233782949975948073200617, 0, 0, 1.33929233782949975948073200617, 1.99796869686072108032830957604, 2.76966127457592119584350767723, 2.94471000703079211711062979119, 3.39983577575754223695834510551, 3.97545217050221942522207015573, 4.25775029536186691313102325812, 4.41998737654410291314220734517, 5.24971375726574680916279888378, 5.63225662549928117828236169746, 6.14932349097873625968388235514, 6.72304875595580026377835859686, 7.01440553160506986767525306197, 7.29304921297364302984044329819, 7.968654378424625052964344745597, 7.975669815462529350073322669559, 8.431635679106594336422474492883, 8.530331281265960957245710013040

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