Properties

Label 2-690-5.3-c2-0-11
Degree $2$
Conductor $690$
Sign $0.998 - 0.0558i$
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 + (−4.92 − 0.875i)5-s − 2.44·6-s + (−4.77 + 4.77i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−5.79 + 4.04i)10-s + 6.21·11-s + (−2.44 + 2.44i)12-s + (−0.0977 − 0.0977i)13-s + 9.54i·14-s + (4.95 + 7.10i)15-s − 4·16-s + (−3.54 + 3.54i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.984 − 0.175i)5-s − 0.408·6-s + (−0.682 + 0.682i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.579 + 0.404i)10-s + 0.565·11-s + (−0.204 + 0.204i)12-s + (−0.00752 − 0.00752i)13-s + 0.682i·14-s + (0.330 + 0.473i)15-s − 0.250·16-s + (−0.208 + 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.998 - 0.0558i$
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: \((2,\ 690,\ (\ :1),\ 0.998 - 0.0558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.247622542\)
\(L(\frac12)\) \(\approx\) \(1.247622542\)
\(L(2)\) 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 \( 1 + (4.92 + 0.875i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (4.77 - 4.77i)T - 49iT^{2} \)
11 \( 1 - 6.21T + 121T^{2} \)
13 \( 1 + (0.0977 + 0.0977i)T + 169iT^{2} \)
17 \( 1 + (3.54 - 3.54i)T - 289iT^{2} \)
19 \( 1 + 5.38iT - 361T^{2} \)
29 \( 1 - 55.0iT - 841T^{2} \)
31 \( 1 - 14.2T + 961T^{2} \)
37 \( 1 + (-41.5 + 41.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 15.3T + 1.68e3T^{2} \)
43 \( 1 + (-40.4 - 40.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.5 + 18.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (30.0 + 30.0i)T + 2.80e3iT^{2} \)
59 \( 1 - 17.7iT - 3.48e3T^{2} \)
61 \( 1 - 104.T + 3.72e3T^{2} \)
67 \( 1 + (50.0 - 50.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 39.0T + 5.04e3T^{2} \)
73 \( 1 + (39.1 + 39.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 71.7iT - 6.24e3T^{2} \)
83 \( 1 + (38.3 + 38.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 20.0iT - 7.92e3T^{2} \)
97 \( 1 + (22.7 - 22.7i)T - 9.40e3iT^{2} \)
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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

−10.58680650400668207139643846984, −9.360706007284167969710043379946, −8.705408091964051017285057623424, −7.49468800985050124273589870662, −6.63217060593772742712049193449, −5.73941682805147284434874488056, −4.69136764309439879031415529476, −3.67905010765857066414904337415, −2.60960570522408032918776963226, −1.01248516274832954785267395569, 0.50773481995379501424649331344, 2.94570141480801629227224427878, 4.03829548640992854999073928661, 4.46031591963032333577309658493, 5.89132341868450902904116194780, 6.66261574742483980318011918894, 7.45600377774775786106941987652, 8.342208708654846786347084191556, 9.456108989437716903650161258701, 10.29300341088013063743094439536

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