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Properties

Label 2-690-5.3-c2-0-4
Degree $2$
Conductor $690$
Sign $-0.232 - 0.972i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (0.0159 + 4.99i)5-s − 2.44·6-s + (−0.165 + 0.165i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (5.01 + 4.98i)10-s + 7.54·11-s + (−2.44 + 2.44i)12-s + (−15.1 − 15.1i)13-s + 0.330i·14-s + (6.10 − 6.14i)15-s − 4·16-s + (−19.2 + 19.2i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.00318 + 0.999i)5-s − 0.408·6-s + (−0.0236 + 0.0236i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.501 + 0.498i)10-s + 0.685·11-s + (−0.204 + 0.204i)12-s + (−1.16 − 1.16i)13-s + 0.0236i·14-s + (0.406 − 0.409i)15-s − 0.250·16-s + (−1.13 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.232 - 0.972i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.232 - 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6331510328\)
\(L(\frac12)\) \(\approx\) \(0.6331510328\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-0.0159 - 4.99i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (0.165 - 0.165i)T - 49iT^{2} \)
11 \( 1 - 7.54T + 121T^{2} \)
13 \( 1 + (15.1 + 15.1i)T + 169iT^{2} \)
17 \( 1 + (19.2 - 19.2i)T - 289iT^{2} \)
19 \( 1 - 11.4iT - 361T^{2} \)
29 \( 1 - 10.0iT - 841T^{2} \)
31 \( 1 + 2.67T + 961T^{2} \)
37 \( 1 + (34.3 - 34.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 29.0T + 1.68e3T^{2} \)
43 \( 1 + (39.6 + 39.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (18.2 - 18.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-14.1 - 14.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 95.0iT - 3.48e3T^{2} \)
61 \( 1 - 21.1T + 3.72e3T^{2} \)
67 \( 1 + (45.5 - 45.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 42.7T + 5.04e3T^{2} \)
73 \( 1 + (-71.8 - 71.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 136. iT - 6.24e3T^{2} \)
83 \( 1 + (7.78 + 7.78i)T + 6.88e3iT^{2} \)
89 \( 1 + 154. iT - 7.92e3T^{2} \)
97 \( 1 + (-35.7 + 35.7i)T - 9.40e3iT^{2} \)
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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

−10.38486529457899113310219283314, −10.24862500324036364553402943108, −8.846973412371275695313300985366, −7.69741987086595319836158078446, −6.80041846168169693477446029853, −6.09179015929060374523318561374, −5.09446003039727210059460264231, −3.86790384191893614943563979103, −2.82353620511547255279164092565, −1.68582002366597946802048282060, 0.18862874392675219083903828612, 2.12534318569336279204925707101, 3.80502249484114846766520247090, 4.78950812932986878102795261451, 5.09873055753396627599483855314, 6.54914106148849975046027610378, 7.05152774206059097316438832259, 8.369258187893982561744663383776, 9.262500610441619710565204344568, 9.652981921848587332997386158435

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