Properties

Label 2-690-5.2-c2-0-15
Degree $2$
Conductor $690$
Sign $0.880 + 0.473i$
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 + (−4.82 − 1.29i)5-s − 2.44·6-s + (5.84 + 5.84i)7-s + (2 − 2i)8-s − 2.99i·9-s + (3.53 + 6.12i)10-s − 1.76·11-s + (2.44 + 2.44i)12-s + (5.00 − 5.00i)13-s − 11.6i·14-s + (−7.49 + 4.33i)15-s − 4·16-s + (−8.50 − 8.50i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.965 − 0.258i)5-s − 0.408·6-s + (0.835 + 0.835i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.353 + 0.612i)10-s − 0.160·11-s + (0.204 + 0.204i)12-s + (0.384 − 0.384i)13-s − 0.835i·14-s + (−0.499 + 0.288i)15-s − 0.250·16-s + (−0.500 − 0.500i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.880 + 0.473i$
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: \((2,\ 690,\ (\ :1),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.435011965\)
\(L(\frac12)\) \(\approx\) \(1.435011965\)
\(L(2)\) 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 \( 1 + (4.82 + 1.29i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (-5.84 - 5.84i)T + 49iT^{2} \)
11 \( 1 + 1.76T + 121T^{2} \)
13 \( 1 + (-5.00 + 5.00i)T - 169iT^{2} \)
17 \( 1 + (8.50 + 8.50i)T + 289iT^{2} \)
19 \( 1 - 22.3iT - 361T^{2} \)
29 \( 1 - 38.8iT - 841T^{2} \)
31 \( 1 - 15.0T + 961T^{2} \)
37 \( 1 + (-28.5 - 28.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 63.9T + 1.68e3T^{2} \)
43 \( 1 + (-34.7 + 34.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-2.22 - 2.22i)T + 2.20e3iT^{2} \)
53 \( 1 + (-67.8 + 67.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 29.3iT - 3.48e3T^{2} \)
61 \( 1 - 54.6T + 3.72e3T^{2} \)
67 \( 1 + (-14.0 - 14.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 36.0T + 5.04e3T^{2} \)
73 \( 1 + (58.7 - 58.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 128. iT - 6.24e3T^{2} \)
83 \( 1 + (68.9 - 68.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 19.7iT - 7.92e3T^{2} \)
97 \( 1 + (-85.9 - 85.9i)T + 9.40e3iT^{2} \)
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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

−10.25324584502597349137229395577, −9.041902772981293716097687647715, −8.480357724622877663436052588875, −7.898289356090858689352264759364, −7.03338682470985999633161696437, −5.63520977869536814653414549387, −4.49455908235700671410634490719, −3.37385024089123325755307946145, −2.25474050819427199350740963235, −0.956525138407758523790452001058, 0.805031059005886703238052068171, 2.55134490662251694645788360682, 4.16008625790592013740482810678, 4.49283114823046538229349536893, 6.00888884353917315709447931124, 7.19884014496412306773260840749, 7.70622717571135387431327118659, 8.515158977448480825157048115499, 9.276169719443709520436100503016, 10.39559722255253249717351340196

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