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Properties

Label 2-690-15.8-c1-0-19
Degree $2$
Conductor $690$
Sign $0.932 + 0.360i$
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  + (−0.707 − 0.707i)2-s + (1.31 − 1.12i)3-s + 1.00i·4-s + (−1.11 + 1.93i)5-s + (−1.72 − 0.138i)6-s + (1.56 − 1.56i)7-s + (0.707 − 0.707i)8-s + (0.477 − 2.96i)9-s + (2.15 − 0.584i)10-s + 5.67i·11-s + (1.12 + 1.31i)12-s + (1.86 + 1.86i)13-s − 2.21·14-s + (0.710 + 3.80i)15-s − 1.00·16-s + (4.87 + 4.87i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.761 − 0.648i)3-s + 0.500i·4-s + (−0.497 + 0.867i)5-s + (−0.704 − 0.0564i)6-s + (0.592 − 0.592i)7-s + (0.250 − 0.250i)8-s + (0.159 − 0.987i)9-s + (0.682 − 0.184i)10-s + 1.71i·11-s + (0.324 + 0.380i)12-s + (0.518 + 0.518i)13-s − 0.592·14-s + (0.183 + 0.983i)15-s − 0.250·16-s + (1.18 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50961 - 0.281760i\)
\(L(\frac12)\) \(\approx\) \(1.50961 - 0.281760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-1.31 + 1.12i)T \)
5 \( 1 + (1.11 - 1.93i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-1.56 + 1.56i)T - 7iT^{2} \)
11 \( 1 - 5.67iT - 11T^{2} \)
13 \( 1 + (-1.86 - 1.86i)T + 13iT^{2} \)
17 \( 1 + (-4.87 - 4.87i)T + 17iT^{2} \)
19 \( 1 + 3.95iT - 19T^{2} \)
29 \( 1 + 0.714T + 29T^{2} \)
31 \( 1 - 6.39T + 31T^{2} \)
37 \( 1 + (2.92 - 2.92i)T - 37iT^{2} \)
41 \( 1 + 4.88iT - 41T^{2} \)
43 \( 1 + (-8.81 - 8.81i)T + 43iT^{2} \)
47 \( 1 + (4.01 + 4.01i)T + 47iT^{2} \)
53 \( 1 + (-5.27 + 5.27i)T - 53iT^{2} \)
59 \( 1 - 7.32T + 59T^{2} \)
61 \( 1 + 4.73T + 61T^{2} \)
67 \( 1 + (-6.02 + 6.02i)T - 67iT^{2} \)
71 \( 1 + 4.27iT - 71T^{2} \)
73 \( 1 + (5.13 + 5.13i)T + 73iT^{2} \)
79 \( 1 + 7.27iT - 79T^{2} \)
83 \( 1 + (10.9 - 10.9i)T - 83iT^{2} \)
89 \( 1 + 8.06T + 89T^{2} \)
97 \( 1 + (0.309 - 0.309i)T - 97iT^{2} \)
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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

−10.31984083561300204961939505474, −9.690273113580281282892807317264, −8.554999378523126584903200864481, −7.78379649002848103000860766132, −7.21576953546394666831306388222, −6.44484690055561370072798066094, −4.47813880775513485240929449174, −3.65152272247068121365449798896, −2.45838553222126574465709106995, −1.38191599776195681264046198094, 1.08127203289847242960743016729, 2.92606913729200132259825177513, 3.99811862125828725026110780769, 5.32439481804803168323712911895, 5.68899205504978740801396311340, 7.47753072714451602223359979142, 8.280362335421247190561700925440, 8.543352723322001988581294726277, 9.385328113584085270814136727704, 10.31208317549853565789080624836

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