Properties

Label 2-690-115.37-c1-0-10
Degree $2$
Conductor $690$
Sign $0.605 + 0.795i$
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 + (−2.21 − 0.326i)5-s + (−0.841 − 0.540i)6-s + (0.672 + 1.80i)7-s + (−0.0713 − 0.997i)8-s + (0.989 + 0.142i)9-s + (1.78 + 1.34i)10-s + (1.31 − 4.48i)11-s + (0.479 + 0.877i)12-s + (−3.13 − 1.16i)13-s + (0.274 − 1.90i)14-s + (−2.18 − 0.483i)15-s + (−0.415 + 0.909i)16-s + (5.05 − 1.09i)17-s + ⋯
L(s)  = 1  + (−0.620 − 0.338i)2-s + (0.575 + 0.0411i)3-s + (0.270 + 0.420i)4-s + (−0.989 − 0.146i)5-s + (−0.343 − 0.220i)6-s + (0.254 + 0.681i)7-s + (−0.0252 − 0.352i)8-s + (0.329 + 0.0474i)9-s + (0.564 + 0.425i)10-s + (0.397 − 1.35i)11-s + (0.138 + 0.253i)12-s + (−0.868 − 0.324i)13-s + (0.0732 − 0.509i)14-s + (−0.563 − 0.124i)15-s + (−0.103 + 0.227i)16-s + (1.22 − 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.605 + 0.795i$
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: \((2,\ 690,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04772 - 0.518949i\)
\(L(\frac12)\) \(\approx\) \(1.04772 - 0.518949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.326i)T \)
23 \( 1 + (-4.76 + 0.568i)T \)
good7 \( 1 + (-0.672 - 1.80i)T + (-5.29 + 4.58i)T^{2} \)
11 \( 1 + (-1.31 + 4.48i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (3.13 + 1.16i)T + (9.82 + 8.51i)T^{2} \)
17 \( 1 + (-5.05 + 1.09i)T + (15.4 - 7.06i)T^{2} \)
19 \( 1 + (1.04 - 0.669i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-4.64 + 7.22i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-0.562 + 0.649i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-5.85 - 7.82i)T + (-10.4 + 35.5i)T^{2} \)
41 \( 1 + (0.209 + 1.45i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.844 + 11.8i)T + (-42.5 - 6.11i)T^{2} \)
47 \( 1 + (-3.86 - 3.86i)T + 47iT^{2} \)
53 \( 1 + (3.54 - 1.32i)T + (40.0 - 34.7i)T^{2} \)
59 \( 1 + (9.22 - 4.21i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (0.718 + 0.622i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-5.62 + 10.2i)T + (-36.2 - 56.3i)T^{2} \)
71 \( 1 + (4.97 - 1.46i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (0.322 - 1.48i)T + (-66.4 - 30.3i)T^{2} \)
79 \( 1 + (-0.824 - 1.80i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (5.47 - 4.09i)T + (23.3 - 79.6i)T^{2} \)
89 \( 1 + (11.7 + 13.5i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-11.8 - 8.84i)T + (27.3 + 93.0i)T^{2} \)
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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

−10.28239736339215837248716076092, −9.338625280870180097839926809549, −8.531897974965408209717567773568, −8.039528770983235999000228870262, −7.21273865382188211533083799178, −5.90450701523279481896025547457, −4.65453390830486304387278720682, −3.42231481047421140266687290031, −2.66968128612933395267765908356, −0.848941403298165831052944140044, 1.28142571708657408134521164907, 2.85927387578595458376568194384, 4.16310326931950911931204917332, 4.93896151623375110728363353177, 6.65720427391513474117567367635, 7.40886775823213461836166335510, 7.72979387651287862701253755195, 8.839410014409066805271727444155, 9.652279163583083441335197700406, 10.41489932224358910447936148413

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