Properties

Label 2-690-15.8-c1-0-34
Degree $2$
Conductor $690$
Sign $0.978 + 0.207i$
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 + (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 + (−1.31 − 1.12i)12-s + (1.86 + 1.86i)13-s + 2.21·14-s + (1.30 + 3.64i)15-s − 1.00·16-s + (−4.87 − 4.87i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.648 + 0.761i)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 + (−0.380 − 0.324i)12-s + (0.518 + 0.518i)13-s + 0.592·14-s + (0.337 + 0.941i)15-s − 0.250·16-s + (−1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.978 + 0.207i$
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: \((2,\ 690,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65740 - 0.174086i\)
\(L(\frac12)\) \(\approx\) \(1.65740 - 0.174086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.93i)T \)
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

−10.70234914210980928291925792386, −9.283889838509397553357672551423, −8.915317985942322521482623136414, −7.86794061748106716680033470217, −6.46841521302527396804903423649, −5.95928259633592805455039915066, −4.72741886886592573358417577514, −4.49978526369227720738454956119, −3.04106181107037466074381587623, −0.842111616070860655519332794420, 1.81013338662361996897990577931, 2.31069871521393836524922403484, 4.02402378140360402103382705104, 5.17091938551355181226733811610, 6.01385789804067746870099514688, 6.73875760804965498178600981599, 7.68728671916604949329171066721, 8.765551654336966525907321765323, 10.13480488843669169256665080620, 10.57013699081323158495784104955

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