Properties

Label 2-690-115.37-c1-0-3
Degree $2$
Conductor $690$
Sign $-0.0187 - 0.999i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (−0.877 − 0.479i)2-s + (−0.997 − 0.0713i)3-s + (0.540 + 0.841i)4-s + (0.565 + 2.16i)5-s + (0.841 + 0.540i)6-s + (1.25 + 3.36i)7-s + (−0.0713 − 0.997i)8-s + (0.989 + 0.142i)9-s + (0.540 − 2.16i)10-s + (0.643 − 2.19i)11-s + (−0.479 − 0.877i)12-s + (2.56 + 0.954i)13-s + (0.511 − 3.55i)14-s + (−0.410 − 2.19i)15-s + (−0.415 + 0.909i)16-s + (0.483 − 0.105i)17-s + ⋯
L(s)  = 1  + (−0.620 − 0.338i)2-s + (−0.575 − 0.0411i)3-s + (0.270 + 0.420i)4-s + (0.253 + 0.967i)5-s + (0.343 + 0.220i)6-s + (0.474 + 1.27i)7-s + (−0.0252 − 0.352i)8-s + (0.329 + 0.0474i)9-s + (0.170 − 0.686i)10-s + (0.194 − 0.660i)11-s + (−0.138 − 0.253i)12-s + (0.710 + 0.264i)13-s + (0.136 − 0.950i)14-s + (−0.105 − 0.567i)15-s + (−0.103 + 0.227i)16-s + (0.117 − 0.0255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0187 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0187 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0187 - 0.999i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.0187 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620048 + 0.631766i\)
\(L(\frac12)\) \(\approx\) \(0.620048 + 0.631766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.877 + 0.479i)T \)
3 \( 1 + (0.997 + 0.0713i)T \)
5 \( 1 + (-0.565 - 2.16i)T \)
23 \( 1 + (3.80 - 2.92i)T \)
good7 \( 1 + (-1.25 - 3.36i)T + (-5.29 + 4.58i)T^{2} \)
11 \( 1 + (-0.643 + 2.19i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.56 - 0.954i)T + (9.82 + 8.51i)T^{2} \)
17 \( 1 + (-0.483 + 0.105i)T + (15.4 - 7.06i)T^{2} \)
19 \( 1 + (2.66 - 1.71i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-1.40 + 2.19i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (2.17 - 2.51i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-2.70 - 3.60i)T + (-10.4 + 35.5i)T^{2} \)
41 \( 1 + (1.05 + 7.32i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (0.0953 - 1.33i)T + (-42.5 - 6.11i)T^{2} \)
47 \( 1 + (-6.96 - 6.96i)T + 47iT^{2} \)
53 \( 1 + (3.48 - 1.29i)T + (40.0 - 34.7i)T^{2} \)
59 \( 1 + (-1.75 + 0.799i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-6.99 - 6.06i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (3.79 - 6.95i)T + (-36.2 - 56.3i)T^{2} \)
71 \( 1 + (8.39 - 2.46i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (2.60 - 11.9i)T + (-66.4 - 30.3i)T^{2} \)
79 \( 1 + (3.20 + 7.00i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (2.35 - 1.76i)T + (23.3 - 79.6i)T^{2} \)
89 \( 1 + (10.3 + 11.8i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-12.4 - 9.34i)T + (27.3 + 93.0i)T^{2} \)
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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

−10.73069303938425386019557543587, −9.973647833454336010989636708224, −8.962012073899262849588354461209, −8.268493543744170761538551713940, −7.19893341053159785899948084186, −6.10173806940208840850873184170, −5.69008480903960964362877047753, −4.02206724820083696835304837390, −2.77700862658722291377089448275, −1.66510933777898173593584064023, 0.64211846914451208911592003932, 1.78668618279241028937564692486, 4.05106071530946155977035771578, 4.76846743134783980610211627438, 5.85752193434652208530277662317, 6.77195204700831857546115955307, 7.70466606376439832437607935656, 8.460089525847497490518518278180, 9.428378941796630101521974634219, 10.27062059754958478547687399390

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