Properties

Label 2-690-5.3-c2-0-3
Degree $2$
Conductor $690$
Sign $-0.954 - 0.299i$
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 + (4.29 + 2.55i)5-s + 2.44·6-s + (4.10 − 4.10i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−6.85 + 1.74i)10-s − 12.9·11-s + (−2.44 + 2.44i)12-s + (−7.48 − 7.48i)13-s + 8.20i·14-s + (−2.13 − 8.39i)15-s − 4·16-s + (−23.2 + 23.2i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.859 + 0.510i)5-s + 0.408·6-s + (0.585 − 0.585i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.685 + 0.174i)10-s − 1.17·11-s + (−0.204 + 0.204i)12-s + (−0.576 − 0.576i)13-s + 0.585i·14-s + (−0.142 − 0.559i)15-s − 0.250·16-s + (−1.36 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.954 - 0.299i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.954 - 0.299i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3614957559\)
\(L(\frac12)\) \(\approx\) \(0.3614957559\)
\(L(2)\) 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 \( 1 + (-4.29 - 2.55i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (-4.10 + 4.10i)T - 49iT^{2} \)
11 \( 1 + 12.9T + 121T^{2} \)
13 \( 1 + (7.48 + 7.48i)T + 169iT^{2} \)
17 \( 1 + (23.2 - 23.2i)T - 289iT^{2} \)
19 \( 1 + 2.66iT - 361T^{2} \)
29 \( 1 - 13.0iT - 841T^{2} \)
31 \( 1 - 21.6T + 961T^{2} \)
37 \( 1 + (1.73 - 1.73i)T - 1.36e3iT^{2} \)
41 \( 1 + 41.1T + 1.68e3T^{2} \)
43 \( 1 + (2.10 + 2.10i)T + 1.84e3iT^{2} \)
47 \( 1 + (2.10 - 2.10i)T - 2.20e3iT^{2} \)
53 \( 1 + (8.58 + 8.58i)T + 2.80e3iT^{2} \)
59 \( 1 - 65.6iT - 3.48e3T^{2} \)
61 \( 1 + 39.1T + 3.72e3T^{2} \)
67 \( 1 + (42.4 - 42.4i)T - 4.48e3iT^{2} \)
71 \( 1 + 119.T + 5.04e3T^{2} \)
73 \( 1 + (86.2 + 86.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 63.9iT - 6.24e3T^{2} \)
83 \( 1 + (-98.1 - 98.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + (-49.2 + 49.2i)T - 9.40e3iT^{2} \)
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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

−10.58387936305119840342963931577, −10.03654159166187760066778013982, −8.762894070853911957911479302795, −7.939108568116480979826080091612, −7.14983792380132542055993206635, −6.32714269256786217309914118993, −5.47674009387560286663955808537, −4.55323921802495064045901284902, −2.65844496480435452647620772612, −1.55168285521832261811422977367, 0.15057510042246014083898488437, 1.90415697811472354436888136128, 2.75354649213804510099118372384, 4.63179174900909655691355018056, 5.04916802517038796650003079647, 6.20382629234574709310267301816, 7.35876850830471669053364335007, 8.481040257616319297447063548699, 9.148272651594609782679924579330, 9.890303284640419436543849745394

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