Properties

Label 2-690-5.2-c2-0-40
Degree $2$
Conductor $690$
Sign $-0.907 + 0.419i$
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 + (3.93 − 3.08i)5-s − 2.44·6-s + (−1.42 − 1.42i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−7.01 − 0.852i)10-s + 3.13·11-s + (2.44 + 2.44i)12-s + (−8.16 + 8.16i)13-s + 2.84i·14-s + (1.04 − 8.59i)15-s − 4·16-s + (−8.80 − 8.80i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.787 − 0.616i)5-s − 0.408·6-s + (−0.203 − 0.203i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.701 − 0.0852i)10-s + 0.284·11-s + (0.204 + 0.204i)12-s + (−0.628 + 0.628i)13-s + 0.203i·14-s + (0.0695 − 0.573i)15-s − 0.250·16-s + (−0.518 − 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.907 + 0.419i$
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: \((2,\ 690,\ (\ :1),\ -0.907 + 0.419i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.380408934\)
\(L(\frac12)\) \(\approx\) \(1.380408934\)
\(L(2)\) 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 \( 1 + (-3.93 + 3.08i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (1.42 + 1.42i)T + 49iT^{2} \)
11 \( 1 - 3.13T + 121T^{2} \)
13 \( 1 + (8.16 - 8.16i)T - 169iT^{2} \)
17 \( 1 + (8.80 + 8.80i)T + 289iT^{2} \)
19 \( 1 + 33.0iT - 361T^{2} \)
29 \( 1 - 8.67iT - 841T^{2} \)
31 \( 1 + 47.5T + 961T^{2} \)
37 \( 1 + (-18.6 - 18.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 21.0T + 1.68e3T^{2} \)
43 \( 1 + (-37.8 + 37.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (4.90 + 4.90i)T + 2.20e3iT^{2} \)
53 \( 1 + (-6.02 + 6.02i)T - 2.80e3iT^{2} \)
59 \( 1 + 91.3iT - 3.48e3T^{2} \)
61 \( 1 + 76.7T + 3.72e3T^{2} \)
67 \( 1 + (-79.6 - 79.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 123.T + 5.04e3T^{2} \)
73 \( 1 + (-89.3 + 89.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 43.8iT - 6.24e3T^{2} \)
83 \( 1 + (72.8 - 72.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 85.1iT - 7.92e3T^{2} \)
97 \( 1 + (104. + 104. i)T + 9.40e3iT^{2} \)
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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

−9.467491383297099633230833025251, −9.307063491929439775626771997254, −8.456687950997212876756443855676, −7.21584038975237767272065018407, −6.66582924864621853469216762801, −5.24323555336108881631092609158, −4.22843176137899461378503945758, −2.76660280229097057413517013064, −1.88434906846765747883875751508, −0.52964387724027835313798538446, 1.71733572419678277086860563824, 2.88922303661857238143718189909, 4.15115676644407558565923949641, 5.59377260147874228443006339909, 6.08865268307343498639957001057, 7.28129288729382779290371157389, 7.993696218870107599302801885218, 9.123200800401250786797491682088, 9.626351069621359028283325563073, 10.44707451487961385999485873524

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