Properties

Label 2-736-184.101-c1-0-11
Degree $2$
Conductor $736$
Sign $0.943 - 0.332i$
Analytic cond. $5.87698$
Root an. cond. $2.42425$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.121 − 0.105i)3-s + (2.18 − 0.314i)5-s + (0.526 − 1.15i)7-s + (−0.423 + 2.94i)9-s + (−1.19 + 4.06i)11-s + (1.87 − 0.856i)13-s + (0.232 − 0.268i)15-s + (6.16 + 3.96i)17-s + (−2.67 − 4.15i)19-s + (−0.0574 − 0.195i)21-s + (3.19 + 3.57i)23-s + (−0.123 + 0.0363i)25-s + (0.519 + 0.808i)27-s + (3.63 − 5.65i)29-s + (−1.04 + 1.20i)31-s + ⋯
L(s)  = 1  + (0.0701 − 0.0608i)3-s + (0.976 − 0.140i)5-s + (0.198 − 0.435i)7-s + (−0.141 + 0.981i)9-s + (−0.359 + 1.22i)11-s + (0.520 − 0.237i)13-s + (0.0600 − 0.0692i)15-s + (1.49 + 0.961i)17-s + (−0.613 − 0.954i)19-s + (−0.0125 − 0.0426i)21-s + (0.666 + 0.745i)23-s + (−0.0247 + 0.00727i)25-s + (0.100 + 0.155i)27-s + (0.674 − 1.04i)29-s + (−0.187 + 0.215i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.943 - 0.332i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87453 + 0.320691i\)
\(L(\frac12)\) \(\approx\) \(1.87453 + 0.320691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (-3.19 - 3.57i)T \)
good3 \( 1 + (-0.121 + 0.105i)T + (0.426 - 2.96i)T^{2} \)
5 \( 1 + (-2.18 + 0.314i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.526 + 1.15i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (1.19 - 4.06i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.87 + 0.856i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-6.16 - 3.96i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (2.67 + 4.15i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-3.63 + 5.65i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (1.04 - 1.20i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (7.32 + 1.05i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.490 + 3.41i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-8.83 + 7.65i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 5.20T + 47T^{2} \)
53 \( 1 + (5.98 + 2.73i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (3.35 - 1.53i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-0.450 - 0.390i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-2.97 - 10.1i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-5.73 + 1.68i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (3.25 - 2.09i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-1.14 - 2.51i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-5.66 - 0.813i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (2.88 + 3.33i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (0.493 + 3.43i)T + (-93.0 + 27.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.40935507159261685342973425591, −9.732997297036052040156711052563, −8.716848456844219729378806800976, −7.77557649523792011710471952132, −7.07042676536713454981172152222, −5.79038562913082996937432878268, −5.16909479568391892818237564087, −4.04000121382866829671376516897, −2.51745582948423884094983682222, −1.51596086841636737394381567690, 1.15055307144828983156681339124, 2.72982794433980306315799424538, 3.58886955921006771043312396408, 5.16065484423969679565001294816, 5.93663598119009135536657719851, 6.53675026364427349011496781532, 7.893557463644341665521498272298, 8.774643198838170635545970477957, 9.400724772027731701571407558957, 10.31505414809108950401615186741

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