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Properties

Label 2-690-15.2-c1-0-39
Degree $2$
Conductor $690$
Sign $-0.761 + 0.648i$
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 + (0.292 − 1.70i)3-s − 1.00i·4-s + (2.12 − 0.707i)5-s + (−0.999 − 1.41i)6-s + (−1 − i)7-s + (−0.707 − 0.707i)8-s + (−2.82 − i)9-s + (0.999 − 2i)10-s + 1.41i·11-s + (−1.70 − 0.292i)12-s + (2 − 2i)13-s − 1.41·14-s + (−0.585 − 3.82i)15-s − 1.00·16-s + (−1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.169 − 0.985i)3-s − 0.500i·4-s + (0.948 − 0.316i)5-s + (−0.408 − 0.577i)6-s + (−0.377 − 0.377i)7-s + (−0.250 − 0.250i)8-s + (−0.942 − 0.333i)9-s + (0.316 − 0.632i)10-s + 0.426i·11-s + (−0.492 − 0.0845i)12-s + (0.554 − 0.554i)13-s − 0.377·14-s + (−0.151 − 0.988i)15-s − 0.250·16-s + (−0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.761 + 0.648i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752369 - 2.04395i\)
\(L(\frac12)\) \(\approx\) \(0.752369 - 2.04395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 + 1.70i)T \)
5 \( 1 + (-2.12 + 0.707i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (2 + 2i)T + 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-9 - 9i)T + 97iT^{2} \)
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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

−10.20483483950884805372379164853, −9.321697536208866898712991900121, −8.518565265666828131075196468765, −7.33277803236962826084809670669, −6.42029835474039863296691124414, −5.75648703466586541607559654382, −4.62420988438161377997862817840, −3.20402780694916566570583405327, −2.18847418686379224649642719106, −0.999242961009195467807474004558, 2.32077793777218274541806795091, 3.39687766395987749385190315893, 4.37598069421156006991121432064, 5.64265338627513340801853088406, 5.96807922949829298289326356963, 7.10624025067158915307790413302, 8.408939297314835834482664498366, 9.134026542282515242186388497653, 9.799845962245753276569630683502, 10.77806616354411218949182848376

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