Properties

Label 2-690-345.89-c1-0-10
Degree $2$
Conductor $690$
Sign $-0.398 - 0.917i$
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 + (0.748 + 1.56i)3-s + (−0.959 + 0.281i)4-s + (−1.64 + 1.51i)5-s + (1.43 − 0.963i)6-s + (1.66 − 1.07i)7-s + (0.415 + 0.909i)8-s + (−1.87 + 2.33i)9-s + (1.73 + 1.41i)10-s + (0.163 − 1.14i)11-s + (−1.15 − 1.28i)12-s + (−3.31 + 5.15i)13-s + (−1.29 − 1.49i)14-s + (−3.59 − 1.43i)15-s + (0.841 − 0.540i)16-s + (−0.599 + 2.04i)17-s + ⋯
L(s)  = 1  + (−0.100 − 0.699i)2-s + (0.432 + 0.901i)3-s + (−0.479 + 0.140i)4-s + (−0.735 + 0.676i)5-s + (0.587 − 0.393i)6-s + (0.629 − 0.404i)7-s + (0.146 + 0.321i)8-s + (−0.626 + 0.779i)9-s + (0.547 + 0.447i)10-s + (0.0494 − 0.343i)11-s + (−0.334 − 0.371i)12-s + (−0.919 + 1.43i)13-s + (−0.346 − 0.399i)14-s + (−0.928 − 0.371i)15-s + (0.210 − 0.135i)16-s + (−0.145 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.398 - 0.917i$
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: \((2,\ 690,\ (\ :1/2),\ -0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505121 + 0.770521i\)
\(L(\frac12)\) \(\approx\) \(0.505121 + 0.770521i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.748 - 1.56i)T \)
5 \( 1 + (1.64 - 1.51i)T \)
23 \( 1 + (4.69 - 0.979i)T \)
good7 \( 1 + (-1.66 + 1.07i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.163 + 1.14i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (3.31 - 5.15i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (0.599 - 2.04i)T + (-14.3 - 9.19i)T^{2} \)
19 \( 1 + (0.290 + 0.988i)T + (-15.9 + 10.2i)T^{2} \)
29 \( 1 + (0.730 - 2.48i)T + (-24.3 - 15.6i)T^{2} \)
31 \( 1 + (2.50 + 5.48i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-5.56 - 6.42i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-2.34 - 2.03i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (5.08 - 11.1i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 4.55T + 47T^{2} \)
53 \( 1 + (-2.42 - 3.76i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (-3.10 + 4.83i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (9.92 - 4.53i)T + (39.9 - 46.1i)T^{2} \)
67 \( 1 + (-0.884 - 6.15i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-12.3 + 1.77i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (0.762 + 2.59i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (-6.33 + 9.85i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (-12.6 + 10.9i)T + (11.8 - 82.1i)T^{2} \)
89 \( 1 + (4.37 - 9.58i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-4.00 + 4.62i)T + (-13.8 - 96.0i)T^{2} \)
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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

−10.83848138404982939827924630520, −9.917334955872145079331602857269, −9.266874148857816693271852899764, −8.176770986526065162675586367490, −7.62231838057227793852817015913, −6.30814293623492138328142079775, −4.69374723548180063183876688823, −4.23102055276758718235994395721, −3.21109880618664452248872857270, −2.04851414367750530769245286287, 0.46372488950608832166696982617, 2.13972474728739290707802860319, 3.63351020504694196613835423002, 4.95521901172778956529012002291, 5.65783904772590100649182386518, 6.95925975746642065562622342447, 7.75238083942178263103992857657, 8.168918313586418915204061478587, 8.976693606947913203932397984566, 9.926737593820243026545716283660

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