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Properties

Label 2-616-11.3-c1-0-5
Degree $2$
Conductor $616$
Sign $-0.147 - 0.989i$
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  + (0.665 + 2.04i)3-s + (2.20 + 1.60i)5-s + (−0.309 + 0.951i)7-s + (−1.32 + 0.962i)9-s + (2.37 + 2.31i)11-s + (0.966 − 0.702i)13-s + (−1.81 + 5.58i)15-s + (−3.00 − 2.18i)17-s + (−0.701 − 2.15i)19-s − 2.15·21-s + 1.70·23-s + (0.751 + 2.31i)25-s + (2.37 + 1.72i)27-s + (1.38 − 4.25i)29-s + (−5.51 + 4.00i)31-s + ⋯
L(s)  = 1  + (0.384 + 1.18i)3-s + (0.986 + 0.716i)5-s + (−0.116 + 0.359i)7-s + (−0.441 + 0.320i)9-s + (0.717 + 0.696i)11-s + (0.268 − 0.194i)13-s + (−0.468 + 1.44i)15-s + (−0.728 − 0.529i)17-s + (−0.160 − 0.495i)19-s − 0.469·21-s + 0.354·23-s + (0.150 + 0.462i)25-s + (0.456 + 0.332i)27-s + (0.257 − 0.791i)29-s + (−0.990 + 0.719i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30217 + 1.51020i\)
\(L(\frac12)\) \(\approx\) \(1.30217 + 1.51020i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 - 2.31i)T \)
good3 \( 1 + (-0.665 - 2.04i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.20 - 1.60i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-0.966 + 0.702i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.00 + 2.18i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.701 + 2.15i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.70T + 23T^{2} \)
29 \( 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.51 - 4.00i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.79 + 5.53i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.488 + 1.50i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.77T + 43T^{2} \)
47 \( 1 + (0.129 + 0.399i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.64 + 1.19i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.39 - 7.36i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.35 + 3.16i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 + (1.26 + 0.918i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.96 - 12.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.7 + 8.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.57 - 5.50i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.67T + 89T^{2} \)
97 \( 1 + (-9.56 + 6.95i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.63310631771328244853608809635, −9.895206103633825619537556286169, −9.316386303744147270749332436522, −8.669441717869679722009520283870, −7.12298918183202790273104590239, −6.38365000892134938788607989660, −5.25370202898430171824920172212, −4.28054051783165983437444087479, −3.17259028702294530491834017962, −2.09407558287129768473336050222, 1.19365066101925706676558967915, 2.01277474078249752843598137209, 3.55040682242966118322683275179, 4.93062608999479818856335364843, 6.17553368541704022478902061621, 6.63913476690318699595997519206, 7.80451680752480914677197293340, 8.656288320573451666801136682192, 9.272371604051189424712416242173, 10.33906043171952564823802785772

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