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Properties

Label 2-690-15.8-c1-0-33
Degree $2$
Conductor $690$
Sign $0.238 + 0.971i$
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 + (1.12 + 1.31i)3-s + 1.00i·4-s + (−0.192 − 2.22i)5-s + (0.133 − 1.72i)6-s + (1.96 − 1.96i)7-s + (0.707 − 0.707i)8-s + (−0.461 + 2.96i)9-s + (−1.43 + 1.71i)10-s − 2.97i·11-s + (−1.31 + 1.12i)12-s + (−1.33 − 1.33i)13-s − 2.77·14-s + (2.71 − 2.76i)15-s − 1.00·16-s + (0.217 + 0.217i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.650 + 0.759i)3-s + 0.500i·4-s + (−0.0860 − 0.996i)5-s + (0.0545 − 0.704i)6-s + (0.740 − 0.740i)7-s + (0.250 − 0.250i)8-s + (−0.153 + 0.988i)9-s + (−0.455 + 0.541i)10-s − 0.896i·11-s + (−0.379 + 0.325i)12-s + (−0.369 − 0.369i)13-s − 0.740·14-s + (0.700 − 0.713i)15-s − 0.250·16-s + (0.0527 + 0.0527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.238 + 0.971i$
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: \((2,\ 690,\ (\ :1/2),\ 0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11392 - 0.873514i\)
\(L(\frac12)\) \(\approx\) \(1.11392 - 0.873514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 1.31i)T \)
5 \( 1 + (0.192 + 2.22i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-1.96 + 1.96i)T - 7iT^{2} \)
11 \( 1 + 2.97iT - 11T^{2} \)
13 \( 1 + (1.33 + 1.33i)T + 13iT^{2} \)
17 \( 1 + (-0.217 - 0.217i)T + 17iT^{2} \)
19 \( 1 + 6.42iT - 19T^{2} \)
29 \( 1 + 1.22T + 29T^{2} \)
31 \( 1 - 6.33T + 31T^{2} \)
37 \( 1 + (-1.43 + 1.43i)T - 37iT^{2} \)
41 \( 1 - 3.04iT - 41T^{2} \)
43 \( 1 + (1.80 + 1.80i)T + 43iT^{2} \)
47 \( 1 + (-0.199 - 0.199i)T + 47iT^{2} \)
53 \( 1 + (-9.17 + 9.17i)T - 53iT^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 - 5.57T + 61T^{2} \)
67 \( 1 + (-9.41 + 9.41i)T - 67iT^{2} \)
71 \( 1 - 7.64iT - 71T^{2} \)
73 \( 1 + (4.05 + 4.05i)T + 73iT^{2} \)
79 \( 1 - 14.4iT - 79T^{2} \)
83 \( 1 + (12.1 - 12.1i)T - 83iT^{2} \)
89 \( 1 + 2.45T + 89T^{2} \)
97 \( 1 + (-8.98 + 8.98i)T - 97iT^{2} \)
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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

−10.13809064981122559070090802091, −9.476133338969256663488531264945, −8.462357002220744278939164563733, −8.226351713551377798987344517858, −7.13832855383238102484937561071, −5.38220049317170786354101667773, −4.58228913159324932196310550497, −3.70538009464117043831560651896, −2.43571767334316011866763236296, −0.854276851751856372026170134688, 1.74455279842357543115424733236, 2.60156116558521086980189893549, 4.09793981528967394637866691174, 5.56876817291574446676879614033, 6.49876712265175828352913734389, 7.29372335806844847473953599549, 7.954859797619476805667238148077, 8.688275041827222897371533422585, 9.732704178279808488545266709473, 10.36200684385697684825866000649

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