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Properties

Label 2-690-5.4-c1-0-17
Degree $2$
Conductor $690$
Sign $-0.970 - 0.241i$
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  − i·2-s − i·3-s − 4-s + (−0.539 + 2.17i)5-s − 6-s − 4.34i·7-s + i·8-s − 9-s + (2.17 + 0.539i)10-s − 1.07·11-s + i·12-s − 2.34i·13-s − 4.34·14-s + (2.17 + 0.539i)15-s + 16-s − 0.921i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.241 + 0.970i)5-s − 0.408·6-s − 1.64i·7-s + 0.353i·8-s − 0.333·9-s + (0.686 + 0.170i)10-s − 0.325·11-s + 0.288i·12-s − 0.649i·13-s − 1.15·14-s + (0.560 + 0.139i)15-s + 0.250·16-s − 0.223i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.970 - 0.241i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0874035 + 0.714246i\)
\(L(\frac12)\) \(\approx\) \(0.0874035 + 0.714246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 + (0.539 - 2.17i)T \)
23 \( 1 - iT \)
good7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 + 1.07T + 11T^{2} \)
13 \( 1 + 2.34iT - 13T^{2} \)
17 \( 1 + 0.921iT - 17T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 2.58iT - 37T^{2} \)
41 \( 1 - 0.156T + 41T^{2} \)
43 \( 1 - 0.738iT - 43T^{2} \)
47 \( 1 + 6.83iT - 47T^{2} \)
53 \( 1 + 0.340iT - 53T^{2} \)
59 \( 1 - 8.83T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 + 2.58iT - 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + 4.68iT - 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 11.4iT - 83T^{2} \)
89 \( 1 - 4.68T + 89T^{2} \)
97 \( 1 + 9.02iT - 97T^{2} \)
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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

−10.33616841259001787373549823414, −9.387348917267846529079730713758, −7.993533162811208663644131946113, −7.46224991788237332139940237311, −6.68335588012595609941406709066, −5.43706490544826856756977452236, −4.01822819301975809525106485614, −3.32651984691583597331848678487, −1.99778208585711164553479391031, −0.36468855572919945059992450275, 2.09291973819645089464573489828, 3.70957367324750481010698043165, 4.79602761902937948653697760210, 5.49288341192927043017928742824, 6.23354506549881304300951990415, 7.64515424915443585291104872232, 8.519688804523526994450849885180, 9.095256774979201630168437382135, 9.607438761040653170181343380545, 10.97745248520765076057331857498

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