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Properties

Label 2-3762-209.208-c1-0-83
Degree $2$
Conductor $3762$
Sign $0.902 + 0.431i$
Analytic cond. $30.0397$
Root an. cond. $5.48085$
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  + 2-s + 4-s + 3.98·5-s + 0.391i·7-s + 8-s + 3.98·10-s + (−2.34 − 2.34i)11-s + 0.870·13-s + 0.391i·14-s + 16-s − 5.35i·17-s + (−4.11 + 1.45i)19-s + 3.98·20-s + (−2.34 − 2.34i)22-s − 2.08·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.78·5-s + 0.148i·7-s + 0.353·8-s + 1.26·10-s + (−0.706 − 0.707i)11-s + 0.241·13-s + 0.104i·14-s + 0.250·16-s − 1.29i·17-s + (−0.942 + 0.332i)19-s + 0.890·20-s + (−0.499 − 0.500i)22-s − 0.435·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3762\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 19\)
Sign: $0.902 + 0.431i$
Analytic conductor: \(30.0397\)
Root analytic conductor: \(5.48085\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3762} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3762,\ (\ :1/2),\ 0.902 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.221886528\)
\(L(\frac12)\) \(\approx\) \(4.221886528\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 + (2.34 + 2.34i)T \)
19 \( 1 + (4.11 - 1.45i)T \)
good5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 - 0.391iT - 7T^{2} \)
13 \( 1 - 0.870T + 13T^{2} \)
17 \( 1 + 5.35iT - 17T^{2} \)
23 \( 1 + 2.08T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 - 5.16iT - 31T^{2} \)
37 \( 1 + 7.83iT - 37T^{2} \)
41 \( 1 - 3.44T + 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 1.51iT - 53T^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 + 0.200iT - 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 4.05iT - 73T^{2} \)
79 \( 1 + 1.63T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 - 16.3iT - 89T^{2} \)
97 \( 1 - 3.49iT - 97T^{2} \)
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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

−8.748372428809376608553196281086, −7.52179646826858107519626421709, −6.73931239460548748811003289830, −6.00226626383631920555276870832, −5.53172189756125725685552399407, −4.92069329107167799446601777192, −3.83265569395249578921401568858, −2.56146279338854123807723425064, −2.38810527591405728927607158963, −0.999449108982013741474382311164, 1.34735217620769386657166357511, 2.23684433528930805076978449124, 2.80090699225777594259582167303, 4.17031182477969605919234704234, 4.78028843035538509701699832277, 5.68735335008663519297338347640, 6.22236527935716039081426813062, 6.70035024860627580125654607706, 7.79604987114322653974324663138, 8.542158956703509139640255907035

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