Properties

Label 2-690-15.14-c2-0-18
Degree $2$
Conductor $690$
Sign $0.901 + 0.433i$
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  − 1.41·2-s + (−2.90 − 0.748i)3-s + 2.00·4-s + (−4.90 − 0.975i)5-s + (4.10 + 1.05i)6-s − 3.09i·7-s − 2.82·8-s + (7.87 + 4.34i)9-s + (6.93 + 1.37i)10-s − 10.9i·11-s + (−5.81 − 1.49i)12-s + 23.5i·13-s + 4.37i·14-s + (13.5 + 6.50i)15-s + 4.00·16-s − 14.9·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.968 − 0.249i)3-s + 0.500·4-s + (−0.980 − 0.195i)5-s + (0.684 + 0.176i)6-s − 0.442i·7-s − 0.353·8-s + (0.875 + 0.483i)9-s + (0.693 + 0.137i)10-s − 0.999i·11-s + (−0.484 − 0.124i)12-s + 1.81i·13-s + 0.312i·14-s + (0.901 + 0.433i)15-s + 0.250·16-s − 0.880·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.433i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.901 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.901 + 0.433i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.901 + 0.433i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4862623250\)
\(L(\frac12)\) \(\approx\) \(0.4862623250\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (2.90 + 0.748i)T \)
5 \( 1 + (4.90 + 0.975i)T \)
23 \( 1 - 4.79T \)
good7 \( 1 + 3.09iT - 49T^{2} \)
11 \( 1 + 10.9iT - 121T^{2} \)
13 \( 1 - 23.5iT - 169T^{2} \)
17 \( 1 + 14.9T + 289T^{2} \)
19 \( 1 + 24.8T + 361T^{2} \)
29 \( 1 - 42.9iT - 841T^{2} \)
31 \( 1 + 31.1T + 961T^{2} \)
37 \( 1 + 48.5iT - 1.36e3T^{2} \)
41 \( 1 - 31.4iT - 1.68e3T^{2} \)
43 \( 1 + 37.4iT - 1.84e3T^{2} \)
47 \( 1 - 50.4T + 2.20e3T^{2} \)
53 \( 1 - 37.0T + 2.80e3T^{2} \)
59 \( 1 - 35.4iT - 3.48e3T^{2} \)
61 \( 1 + 17.9T + 3.72e3T^{2} \)
67 \( 1 - 68.1iT - 4.48e3T^{2} \)
71 \( 1 + 99.8iT - 5.04e3T^{2} \)
73 \( 1 + 39.3iT - 5.32e3T^{2} \)
79 \( 1 - 30.7T + 6.24e3T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 + 146. iT - 7.92e3T^{2} \)
97 \( 1 + 118. iT - 9.40e3T^{2} \)
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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

−10.67223690643592586733112365649, −9.053968275316461612805010414607, −8.685867593364344728273882681236, −7.33424650049014786350808454948, −6.93175840956600083950497268961, −5.96211093395760375166297504358, −4.61099064975964976831761316817, −3.81582658877195860064390563632, −1.90853909182492098670274379937, −0.51456903204528775762608028769, 0.55493874570218446418645521510, 2.39213581543491286609506805088, 3.87584519808839653896895332442, 4.88046934525909020744222657773, 5.97880207543366065571053848409, 6.89085790505577267545006887494, 7.72380767520921388880269221432, 8.540618382965249320693588358867, 9.621387499933651893626590279875, 10.52726260908855229992472195482

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