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Properties

Label 2-690-115.114-c2-0-13
Degree $2$
Conductor $690$
Sign $0.950 + 0.309i$
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  − 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (−4.95 + 0.671i)5-s + 2.44·6-s − 12.0·7-s + 2.82i·8-s − 2.99·9-s + (0.949 + 7.00i)10-s − 20.5i·11-s − 3.46i·12-s + 13.6i·13-s + 17.1i·14-s + (−1.16 − 8.58i)15-s + 4.00·16-s − 0.319·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (−0.990 + 0.134i)5-s + 0.408·6-s − 1.72·7-s + 0.353i·8-s − 0.333·9-s + (0.0949 + 0.700i)10-s − 1.86i·11-s − 0.288i·12-s + 1.05i·13-s + 1.22i·14-s + (−0.0775 − 0.572i)15-s + 0.250·16-s − 0.0187·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.950 + 0.309i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.950 + 0.309i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8023381501\)
\(L(\frac12)\) \(\approx\) \(0.8023381501\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 - 1.73iT \)
5 \( 1 + (4.95 - 0.671i)T \)
23 \( 1 + (-20.7 - 9.98i)T \)
good7 \( 1 + 12.0T + 49T^{2} \)
11 \( 1 + 20.5iT - 121T^{2} \)
13 \( 1 - 13.6iT - 169T^{2} \)
17 \( 1 + 0.319T + 289T^{2} \)
19 \( 1 - 6.05iT - 361T^{2} \)
29 \( 1 + 6.76T + 841T^{2} \)
31 \( 1 + 30.6T + 961T^{2} \)
37 \( 1 - 71.2T + 1.36e3T^{2} \)
41 \( 1 - 12.7T + 1.68e3T^{2} \)
43 \( 1 + 0.0776T + 1.84e3T^{2} \)
47 \( 1 - 35.9iT - 2.20e3T^{2} \)
53 \( 1 + 11.2T + 2.80e3T^{2} \)
59 \( 1 - 47.3T + 3.48e3T^{2} \)
61 \( 1 - 42.0iT - 3.72e3T^{2} \)
67 \( 1 + 75.9T + 4.48e3T^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 + 107. iT - 5.32e3T^{2} \)
79 \( 1 + 68.2iT - 6.24e3T^{2} \)
83 \( 1 + 103.T + 6.88e3T^{2} \)
89 \( 1 - 50.8iT - 7.92e3T^{2} \)
97 \( 1 + 39.8T + 9.40e3T^{2} \)
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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

−10.33552902354862257725755181698, −9.244315199687459395520763075250, −8.962837059144720743901800552125, −7.75119773082532959897305101337, −6.55522302818351141238151802095, −5.71836831808071955441135701443, −4.25895693313149407497199406073, −3.47805678496845650712345792007, −2.92162548166485279663361244849, −0.60516252882422781502465058013, 0.56049997786243895808952571425, 2.70773657279989800207143162191, 3.80664562519918322954586091546, 4.90344148233462557595183878632, 6.06627546828880675971268189449, 7.13556376144959815078347677954, 7.27260984308836359622190197045, 8.409974186666422633528377676624, 9.407819471907592389444754859612, 10.03329352715950395648348028525

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