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Properties

Label 2-690-115.22-c1-0-10
Degree $2$
Conductor $690$
Sign $0.998 + 0.0553i$
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 + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−1.07 + 1.95i)5-s + 1.00·6-s + (3.43 − 3.43i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (2.14 − 0.624i)10-s + 1.77i·11-s + (−0.707 − 0.707i)12-s + (1.21 − 1.21i)13-s − 4.85·14-s + (−0.624 − 2.14i)15-s − 1.00·16-s + (−1.12 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.481 + 0.876i)5-s + 0.408·6-s + (1.29 − 1.29i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.678 − 0.197i)10-s + 0.536i·11-s + (−0.204 − 0.204i)12-s + (0.337 − 0.337i)13-s − 1.29·14-s + (−0.161 − 0.554i)15-s − 0.250·16-s + (−0.272 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.998 + 0.0553i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.998 + 0.0553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07105 - 0.0296699i\)
\(L(\frac12)\) \(\approx\) \(1.07105 - 0.0296699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.07 - 1.95i)T \)
23 \( 1 + (-4.76 - 0.510i)T \)
good7 \( 1 + (-3.43 + 3.43i)T - 7iT^{2} \)
11 \( 1 - 1.77iT - 11T^{2} \)
13 \( 1 + (-1.21 + 1.21i)T - 13iT^{2} \)
17 \( 1 + (1.12 - 1.12i)T - 17iT^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
29 \( 1 + 3.23iT - 29T^{2} \)
31 \( 1 - 6.41T + 31T^{2} \)
37 \( 1 + (-0.321 + 0.321i)T - 37iT^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + (-6.47 - 6.47i)T + 43iT^{2} \)
47 \( 1 + (-9.10 - 9.10i)T + 47iT^{2} \)
53 \( 1 + (-6.42 - 6.42i)T + 53iT^{2} \)
59 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 + (-0.586 + 0.586i)T - 67iT^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + (-3.84 + 3.84i)T - 73iT^{2} \)
79 \( 1 + 6.55T + 79T^{2} \)
83 \( 1 + (2.56 + 2.56i)T + 83iT^{2} \)
89 \( 1 + 3.44T + 89T^{2} \)
97 \( 1 + (-7.79 + 7.79i)T - 97iT^{2} \)
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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

−10.69777159863644583719850966530, −9.962033917815709290879404029338, −8.751854153696338708832276329461, −7.72510649938449479924386909403, −7.29278829321260618076913263324, −6.11245411389082181553897142635, −4.47531549852376221533279790218, −4.12978247369033949632575904558, −2.67308099811598484422459727799, −1.03475265643348483738321582965, 1.00689617252069077889543979487, 2.30497590542908274437962396877, 4.36548876603513812534289775751, 5.25316657376776991571476171507, 5.88302823852767019684406375241, 7.09328738143492498042550881665, 8.055318787790011873865039212837, 8.731914649173048843811457106418, 9.070465795609330425207189420360, 10.65755125203145263528157560765

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