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Properties

Label 4-1101e2-1.1-c1e2-0-0
Degree $4$
Conductor $1212201$
Sign $1$
Analytic cond. $77.2909$
Root an. cond. $2.96505$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 10·7-s − 3·9-s + 2·13-s + 15·19-s − 10·25-s + 20·28-s − 11·31-s + 6·36-s + 37-s − 3·43-s + 61·49-s − 4·52-s − 61-s + 30·63-s + 8·64-s + 11·67-s + 10·73-s − 30·76-s − 30·79-s + 9·81-s − 20·91-s − 33·97-s + 20·100-s − 33·103-s − 6·117-s + 11·121-s + ⋯
L(s)  = 1  − 4-s − 3.77·7-s − 9-s + 0.554·13-s + 3.44·19-s − 2·25-s + 3.77·28-s − 1.97·31-s + 36-s + 0.164·37-s − 0.457·43-s + 61/7·49-s − 0.554·52-s − 0.128·61-s + 3.77·63-s + 64-s + 1.34·67-s + 1.17·73-s − 3.44·76-s − 3.37·79-s + 81-s − 2.09·91-s − 3.35·97-s + 2·100-s − 3.25·103-s − 0.554·117-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1212201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1212201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1212201\)    =    \(3^{2} \cdot 367^{2}\)
Sign: $1$
Analytic conductor: \(77.2909\)
Root analytic conductor: \(2.96505\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1212201,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + p T^{2} \)
367$C_2$ \( 1 + 35 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.7.k_bn
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.19.ap_dq
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.l_dm
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.41.a_bp
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.d_bu
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.al_cc
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ak_bb
79$C_2$ \( ( 1 + 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.be_op
83$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.83.a_df
89$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.89.a_dl
97$C_2$ \( ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.bh_rs
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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

−7.27976356210723365170189569376, −7.26155877756811501087228708745, −6.71114593337755081526564582399, −6.13027085184381531415104668699, −5.84477945575599569033934225657, −5.39521909854607518216988586460, −5.25925738046771771539722303077, −3.94910170803881289686748638135, −3.66829638301627508503104266936, −3.61484332739913987643154053630, −2.77200370882739367660359438765, −2.72553058576893774849951458652, −1.18497641861601885699543675300, 0, 0, 1.18497641861601885699543675300, 2.72553058576893774849951458652, 2.77200370882739367660359438765, 3.61484332739913987643154053630, 3.66829638301627508503104266936, 3.94910170803881289686748638135, 5.25925738046771771539722303077, 5.39521909854607518216988586460, 5.84477945575599569033934225657, 6.13027085184381531415104668699, 6.71114593337755081526564582399, 7.26155877756811501087228708745, 7.27976356210723365170189569376

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