Properties

Label 2-690-23.4-c1-0-5
Degree $2$
Conductor $690$
Sign $-0.365 - 0.930i$
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  + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (0.654 + 0.755i)6-s + (0.455 + 3.16i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (−1.70 + 3.74i)11-s + (−0.415 + 0.909i)12-s + (0.401 − 2.79i)13-s + (−2.69 + 1.72i)14-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (5.33 + 6.15i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.376 − 0.241i)5-s + (0.267 + 0.308i)6-s + (0.172 + 1.19i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0450 − 0.313i)10-s + (−0.515 + 1.12i)11-s + (−0.119 + 0.262i)12-s + (0.111 − 0.774i)13-s + (−0.719 + 0.462i)14-s + (−0.247 − 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (1.29 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.365 - 0.930i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990142 + 1.45286i\)
\(L(\frac12)\) \(\approx\) \(0.990142 + 1.45286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
3 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (4.79 + 0.138i)T \)
good7 \( 1 + (-0.455 - 3.16i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (1.70 - 3.74i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.401 + 2.79i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-5.33 - 6.15i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.54 - 1.78i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-2.60 - 3.00i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (1.90 + 0.558i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-1.51 + 0.970i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-2.69 - 1.73i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-5.02 + 1.47i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 6.43T + 47T^{2} \)
53 \( 1 + (-0.655 - 4.55i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (0.277 - 1.92i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-10.9 - 3.21i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (2.01 + 4.40i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (4.33 + 9.48i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-8.35 + 9.63i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.92 + 13.3i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (3.43 - 2.20i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (10.2 - 3.00i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (10.0 + 6.47i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.54116531774222743411882280120, −9.745381935606385467936001320084, −8.684960000537602319890793837740, −8.057009848949325381925431266160, −7.51692757113286276938240360384, −6.13792924450717741655266997160, −5.46078578036087336937972203420, −4.34228307965916640506462297435, −3.23527348863993454173250606917, −1.93131047533423196594365687610, 0.821371160515337860463081977098, 2.57467066943881102907481858182, 3.57262881499687628580145265555, 4.32199629315187596088941089113, 5.44932134833782619879349772887, 6.78332225240443220091481683097, 7.70352036434726436169371850158, 8.451938416549158402602243916230, 9.618035835811879990560917691124, 10.19951757082769197262097657642

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