↑

Properties

Label 4-2312e2-1.1-c1e2-0-3
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·9-s + 8·13-s − 16·19-s + 8·25-s − 16·43-s + 6·49-s + 24·53-s − 16·59-s + 24·67-s − 5·81-s + 16·89-s − 24·101-s + 16·103-s − 16·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 32·171-s + 173-s + ⋯
L(s)  = 1  − 2/3·9-s + 2.21·13-s − 3.67·19-s + 8/5·25-s − 2.43·43-s + 6/7·49-s + 3.29·53-s − 2.08·59-s + 2.93·67-s − 5/9·81-s + 1.69·89-s − 2.38·101-s + 1.57·103-s − 1.47·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 2.44·171-s + 0.0760·173-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: \(0\)
Selberg data: \((4,\ 5345344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.885619007\)
\(L(\frac12)\) \(\approx\) \(1.885619007\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.23.a_abm
29$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.29.a_abo
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.37.a_ay
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.41.a_adc
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \) 2.61.a_aeq
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.71.a_acs
73$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \) 2.73.a_aey
79$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.79.a_afu
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.89.aq_ji
97$C_2^2$ \( 1 - 176 T^{2} + p^{2} T^{4} \) 2.97.a_agu
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.445574525863765866635981482420, −8.692279094981093538957414678098, −8.493464909765807155342515958165, −8.328043564344854417586216527428, −7.86953210115267914392185915874, −6.95932569511754685355581810648, −6.89932973446684652899001487050, −6.34889549183249876808923724279, −6.28130595910653745491946988123, −5.69389425016615310202932764507, −5.37842065511028981237888268256, −4.66426540231248250444074119182, −4.41068115579316540703722703384, −3.84628160144947740861613983374, −3.56380143762130036695747000730, −3.02829970710837481977952053316, −2.27096435274525188889760291453, −2.05341851123593476570882840595, −1.24015693074733944844306163768, −0.49711944476743301157596755531, 0.49711944476743301157596755531, 1.24015693074733944844306163768, 2.05341851123593476570882840595, 2.27096435274525188889760291453, 3.02829970710837481977952053316, 3.56380143762130036695747000730, 3.84628160144947740861613983374, 4.41068115579316540703722703384, 4.66426540231248250444074119182, 5.37842065511028981237888268256, 5.69389425016615310202932764507, 6.28130595910653745491946988123, 6.34889549183249876808923724279, 6.89932973446684652899001487050, 6.95932569511754685355581810648, 7.86953210115267914392185915874, 8.328043564344854417586216527428, 8.493464909765807155342515958165, 8.692279094981093538957414678098, 9.445574525863765866635981482420

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