Properties

Label 2-736-184.101-c1-0-13
Degree $2$
Conductor $736$
Sign $0.977 - 0.209i$
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  + (−1.40 + 1.21i)3-s + (2.40 − 0.346i)5-s + (−0.345 + 0.756i)7-s + (0.0626 − 0.435i)9-s + (1.73 − 5.91i)11-s + (4.96 − 2.26i)13-s + (−2.95 + 3.40i)15-s + (−5.18 − 3.33i)17-s + (2.95 + 4.59i)19-s + (−0.434 − 1.47i)21-s + (4.45 + 1.76i)23-s + (0.875 − 0.256i)25-s + (−2.56 − 3.99i)27-s + (−0.212 + 0.329i)29-s + (2.66 − 3.07i)31-s + ⋯
L(s)  = 1  + (−0.809 + 0.701i)3-s + (1.07 − 0.154i)5-s + (−0.130 + 0.285i)7-s + (0.0208 − 0.145i)9-s + (0.523 − 1.78i)11-s + (1.37 − 0.628i)13-s + (−0.762 + 0.879i)15-s + (−1.25 − 0.808i)17-s + (0.677 + 1.05i)19-s + (−0.0947 − 0.322i)21-s + (0.929 + 0.368i)23-s + (0.175 − 0.0513i)25-s + (−0.494 − 0.768i)27-s + (−0.0393 + 0.0612i)29-s + (0.478 − 0.552i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.977 - 0.209i$
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: \((2,\ 736,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46393 + 0.155120i\)
\(L(\frac12)\) \(\approx\) \(1.46393 + 0.155120i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.45 - 1.76i)T \)
good3 \( 1 + (1.40 - 1.21i)T + (0.426 - 2.96i)T^{2} \)
5 \( 1 + (-2.40 + 0.346i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.345 - 0.756i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-1.73 + 5.91i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-4.96 + 2.26i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (5.18 + 3.33i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-2.95 - 4.59i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (0.212 - 0.329i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.66 + 3.07i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-6.22 - 0.895i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (-0.756 - 5.25i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.14 + 0.988i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 4.29T + 47T^{2} \)
53 \( 1 + (6.44 + 2.94i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (0.903 - 0.412i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-5.37 - 4.66i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-0.823 - 2.80i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (0.941 - 0.276i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (1.63 - 1.05i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-2.16 - 4.73i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-4.46 - 0.642i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-2.14 - 2.47i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.28 + 8.94i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.52454709367348992665009441031, −9.559518565585496660879068672731, −8.933729558998921058261883158137, −8.016196438063665660353887842709, −6.37672840745488693478241620354, −5.87369214379916680692500645883, −5.31407855086733432544962796915, −4.00715801169966911502665407414, −2.83722608827551176529731614026, −1.06434559096148379668356544654, 1.25346627473211901701335176082, 2.23866046295659321490926807641, 4.01733648195211620127111480436, 5.05201366819607991974059465051, 6.32698492383357970261797500244, 6.55549778646684726485499823886, 7.35973041564773908601230780344, 8.940166551764810034805197102310, 9.397689425491940772950886509515, 10.48785114947615986442678613834

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