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Properties

Label 2-616-11.4-c1-0-7
Degree $2$
Conductor $616$
Sign $0.281 - 0.959i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
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  + (−0.645 + 1.98i)3-s + (1.70 − 1.24i)5-s + (−0.309 − 0.951i)7-s + (−1.10 − 0.801i)9-s + (−2.67 + 1.96i)11-s + (4.98 + 3.62i)13-s + (1.36 + 4.19i)15-s + (2.23 − 1.62i)17-s + (−0.142 + 0.438i)19-s + 2.08·21-s + 6.31·23-s + (−0.165 + 0.508i)25-s + (−2.76 + 2.00i)27-s + (0.693 + 2.13i)29-s + (−2.67 − 1.94i)31-s + ⋯
L(s)  = 1  + (−0.372 + 1.14i)3-s + (0.764 − 0.555i)5-s + (−0.116 − 0.359i)7-s + (−0.367 − 0.267i)9-s + (−0.805 + 0.592i)11-s + (1.38 + 1.00i)13-s + (0.352 + 1.08i)15-s + (0.541 − 0.393i)17-s + (−0.0326 + 0.100i)19-s + 0.455·21-s + 1.31·23-s + (−0.0330 + 0.101i)25-s + (−0.532 + 0.386i)27-s + (0.128 + 0.396i)29-s + (−0.480 − 0.349i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ 0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19364 + 0.894068i\)
\(L(\frac12)\) \(\approx\) \(1.19364 + 0.894068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.67 - 1.96i)T \)
good3 \( 1 + (0.645 - 1.98i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.70 + 1.24i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-4.98 - 3.62i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.23 + 1.62i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.142 - 0.438i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.31T + 23T^{2} \)
29 \( 1 + (-0.693 - 2.13i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.67 + 1.94i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.26 - 6.95i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.0315 + 0.0970i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.33T + 43T^{2} \)
47 \( 1 + (2.56 - 7.88i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.1 + 7.37i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.636 - 1.95i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.15 - 1.56i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 + (-11.4 + 8.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.651 + 2.00i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.93 - 1.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.49 + 4.72i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + (9.99 + 7.26i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.87409295717971019822567427717, −9.692302646732514764244419132142, −9.505418521297275640777555345690, −8.412982415576980743644359736857, −7.18228643219678665725729532878, −6.05789220271712297991806820819, −5.11924523453322672450319476154, −4.47783737305801165923088126964, −3.29852009936202648505077446282, −1.53160812047034509785150573316, 0.979011188043482530506362601443, 2.37551736154765562121957564007, 3.45602401570214832719401438291, 5.42566359333246566965838394674, 5.97111487880539918218884753873, 6.71064208366805988982927567297, 7.75937103700763838288745068622, 8.495086459075226834662463790238, 9.644783790392192331333720186226, 10.73345042976541322331182293827

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