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Properties

Label 2-690-5.3-c2-0-34
Degree $2$
Conductor $690$
Sign $-0.926 + 0.375i$
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 + (−4.94 + 0.763i)5-s + 2.44·6-s + (8.62 − 8.62i)7-s + (2 + 2i)8-s + 2.99i·9-s + (4.17 − 5.70i)10-s − 9.96·11-s + (−2.44 + 2.44i)12-s + (−1.36 − 1.36i)13-s + 17.2i·14-s + (6.98 + 5.11i)15-s − 4·16-s + (12.8 − 12.8i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.988 + 0.152i)5-s + 0.408·6-s + (1.23 − 1.23i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.417 − 0.570i)10-s − 0.905·11-s + (−0.204 + 0.204i)12-s + (−0.105 − 0.105i)13-s + 1.23i·14-s + (0.465 + 0.341i)15-s − 0.250·16-s + (0.756 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.926 + 0.375i$
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: \((2,\ 690,\ (\ :1),\ -0.926 + 0.375i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4062905249\)
\(L(\frac12)\) \(\approx\) \(0.4062905249\)
\(L(2)\) 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 \( 1 + (4.94 - 0.763i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (-8.62 + 8.62i)T - 49iT^{2} \)
11 \( 1 + 9.96T + 121T^{2} \)
13 \( 1 + (1.36 + 1.36i)T + 169iT^{2} \)
17 \( 1 + (-12.8 + 12.8i)T - 289iT^{2} \)
19 \( 1 + 21.6iT - 361T^{2} \)
29 \( 1 - 41.0iT - 841T^{2} \)
31 \( 1 - 39.6T + 961T^{2} \)
37 \( 1 + (47.8 - 47.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 46.0T + 1.68e3T^{2} \)
43 \( 1 + (42.5 + 42.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-4.44 + 4.44i)T - 2.20e3iT^{2} \)
53 \( 1 + (10.1 + 10.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 83.8iT - 3.48e3T^{2} \)
61 \( 1 + 94.9T + 3.72e3T^{2} \)
67 \( 1 + (24.0 - 24.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 127.T + 5.04e3T^{2} \)
73 \( 1 + (-59.6 - 59.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 27.1iT - 6.24e3T^{2} \)
83 \( 1 + (61.0 + 61.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 84.3iT - 7.92e3T^{2} \)
97 \( 1 + (111. - 111. i)T - 9.40e3iT^{2} \)
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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

−10.18250448997835731132089727460, −8.612365419254818709490871479324, −8.005901212480507804427893673797, −7.24872307061706021496418774611, −6.84638309148774571287842168912, −5.06554680790998026271051157908, −4.79496011461486849600102892865, −3.17690801277998544881128532380, −1.33165710674930148307775378351, −0.19984705066859723583288166904, 1.58628127748285557232341401433, 2.94493690283692212552009887558, 4.19471225984629355437780776882, 5.08552746964969975222245061685, 5.99682838994412596301781987322, 7.61832817292469353656067717996, 8.181895140674864207937747930688, 8.754362956599752610559633452607, 9.975513435096560575886517196602, 10.68088010486678158961349758297

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