Properties

Label 2-690-345.89-c1-0-4
Degree $2$
Conductor $690$
Sign $0.632 - 0.774i$
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  + (−0.142 − 0.989i)2-s + (−0.397 − 1.68i)3-s + (−0.959 + 0.281i)4-s + (2.02 + 0.949i)5-s + (−1.61 + 0.633i)6-s + (−4.08 + 2.62i)7-s + (0.415 + 0.909i)8-s + (−2.68 + 1.33i)9-s + (0.651 − 2.13i)10-s + (0.475 − 3.30i)11-s + (0.856 + 1.50i)12-s + (−0.660 + 1.02i)13-s + (3.17 + 3.66i)14-s + (0.796 − 3.79i)15-s + (0.841 − 0.540i)16-s + (−1.75 + 5.96i)17-s + ⋯
L(s)  = 1  + (−0.100 − 0.699i)2-s + (−0.229 − 0.973i)3-s + (−0.479 + 0.140i)4-s + (0.905 + 0.424i)5-s + (−0.658 + 0.258i)6-s + (−1.54 + 0.991i)7-s + (0.146 + 0.321i)8-s + (−0.894 + 0.446i)9-s + (0.206 − 0.676i)10-s + (0.143 − 0.996i)11-s + (0.247 + 0.434i)12-s + (−0.183 + 0.285i)13-s + (0.849 + 0.980i)14-s + (0.205 − 0.978i)15-s + (0.210 − 0.135i)16-s + (−0.424 + 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567695 + 0.269276i\)
\(L(\frac12)\) \(\approx\) \(0.567695 + 0.269276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + (0.397 + 1.68i)T \)
5 \( 1 + (-2.02 - 0.949i)T \)
23 \( 1 + (4.18 - 2.33i)T \)
good7 \( 1 + (4.08 - 2.62i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.475 + 3.30i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.660 - 1.02i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (1.75 - 5.96i)T + (-14.3 - 9.19i)T^{2} \)
19 \( 1 + (-0.670 - 2.28i)T + (-15.9 + 10.2i)T^{2} \)
29 \( 1 + (1.83 - 6.25i)T + (-24.3 - 15.6i)T^{2} \)
31 \( 1 + (2.18 + 4.79i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-0.549 - 0.633i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-4.44 - 3.84i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-0.0334 + 0.0733i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 4.46T + 47T^{2} \)
53 \( 1 + (5.83 + 9.08i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (2.51 - 3.90i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (9.97 - 4.55i)T + (39.9 - 46.1i)T^{2} \)
67 \( 1 + (0.376 + 2.62i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-8.42 + 1.21i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-0.938 - 3.19i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (3.34 - 5.19i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (10.3 - 8.97i)T + (11.8 - 82.1i)T^{2} \)
89 \( 1 + (-2.16 + 4.74i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (4.90 - 5.65i)T + (-13.8 - 96.0i)T^{2} \)
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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

−10.67006837846341537533394602624, −9.703932778685052659718349941651, −9.041654433767147580542116148783, −8.179718553733029127135716433838, −6.82668978128160077324342246439, −5.97968666311132278308849659103, −5.70122461892748829003144119554, −3.58833756429140106230410771677, −2.68572998867120167647623879112, −1.69318240327094258799445609369, 0.33712968901743576905729241919, 2.77056985018142941935369465327, 4.12426391943577979780921902898, 4.86190618812593573476226360068, 5.93569272769273618828669358825, 6.67115633574720538924306828844, 7.53774522085541973306109169775, 9.089073967651144374679815044677, 9.478362545651187985792715740788, 10.03792637885312035455740526359

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