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Properties

Label 2-690-5.2-c2-0-7
Degree $2$
Conductor $690$
Sign $0.385 - 0.922i$
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 + (0.815 − 4.93i)5-s − 2.44·6-s + (5.39 + 5.39i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−5.74 + 4.11i)10-s − 16.8·11-s + (2.44 + 2.44i)12-s + (−15.3 + 15.3i)13-s − 10.7i·14-s + (−5.04 − 7.04i)15-s − 4·16-s + (20.2 + 20.2i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.163 − 0.986i)5-s − 0.408·6-s + (0.771 + 0.771i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.574 + 0.411i)10-s − 1.53·11-s + (0.204 + 0.204i)12-s + (−1.17 + 1.17i)13-s − 0.771i·14-s + (−0.336 − 0.469i)15-s − 0.250·16-s + (1.19 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.385 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8290909102\)
\(L(\frac12)\) \(\approx\) \(0.8290909102\)
\(L(2)\) 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 \( 1 + (-0.815 + 4.93i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (-5.39 - 5.39i)T + 49iT^{2} \)
11 \( 1 + 16.8T + 121T^{2} \)
13 \( 1 + (15.3 - 15.3i)T - 169iT^{2} \)
17 \( 1 + (-20.2 - 20.2i)T + 289iT^{2} \)
19 \( 1 - 27.2iT - 361T^{2} \)
29 \( 1 + 1.86iT - 841T^{2} \)
31 \( 1 + 56.4T + 961T^{2} \)
37 \( 1 + (-14.7 - 14.7i)T + 1.36e3iT^{2} \)
41 \( 1 + 31.1T + 1.68e3T^{2} \)
43 \( 1 + (-3.71 + 3.71i)T - 1.84e3iT^{2} \)
47 \( 1 + (52.2 + 52.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (29.1 - 29.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 44.5iT - 3.48e3T^{2} \)
61 \( 1 - 97.7T + 3.72e3T^{2} \)
67 \( 1 + (-29.7 - 29.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 112.T + 5.04e3T^{2} \)
73 \( 1 + (69.1 - 69.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 17.0iT - 6.24e3T^{2} \)
83 \( 1 + (-70.4 + 70.4i)T - 6.88e3iT^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (78.1 + 78.1i)T + 9.40e3iT^{2} \)
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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

−10.14782894849648174826108438262, −9.636921499498462088854068027548, −8.490396977148700019641858159562, −8.158891788344957762170935959445, −7.35310920615138678682182453285, −5.73057621407843737244155650045, −5.04057879588942397436286278978, −3.73481407123118000600141298986, −2.19779937866165405632101206830, −1.62608490718976798519022694095, 0.31022779458824529116733912108, 2.37304949203816091890234665562, 3.23274399684737211487407673691, 4.96487049993800463939370241641, 5.34575527501025243607304791432, 6.97320348558480840820071242350, 7.67913793164934204041227364471, 7.948406193736448639586027361459, 9.475584564184020345095951505847, 10.02085063406732237879576266331

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