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Properties

Label 2-690-15.8-c1-0-22
Degree $2$
Conductor $690$
Sign $0.0303 - 0.999i$
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 + (1.66 + 0.471i)3-s + 1.00i·4-s + (1.55 − 1.60i)5-s + (0.844 + 1.51i)6-s + (−3.44 + 3.44i)7-s + (−0.707 + 0.707i)8-s + (2.55 + 1.57i)9-s + (2.23 − 0.0307i)10-s + 3.84i·11-s + (−0.471 + 1.66i)12-s + (0.875 + 0.875i)13-s − 4.87·14-s + (3.35 − 1.93i)15-s − 1.00·16-s + (−1.35 − 1.35i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.962 + 0.272i)3-s + 0.500i·4-s + (0.697 − 0.716i)5-s + (0.344 + 0.617i)6-s + (−1.30 + 1.30i)7-s + (−0.250 + 0.250i)8-s + (0.851 + 0.524i)9-s + (0.707 − 0.00972i)10-s + 1.15i·11-s + (−0.136 + 0.481i)12-s + (0.242 + 0.242i)13-s − 1.30·14-s + (0.866 − 0.499i)15-s − 0.250·16-s + (−0.329 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0303 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0303 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0303 - 0.999i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.0303 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89838 + 1.84157i\)
\(L(\frac12)\) \(\approx\) \(1.89838 + 1.84157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 0.471i)T \)
5 \( 1 + (-1.55 + 1.60i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (3.44 - 3.44i)T - 7iT^{2} \)
11 \( 1 - 3.84iT - 11T^{2} \)
13 \( 1 + (-0.875 - 0.875i)T + 13iT^{2} \)
17 \( 1 + (1.35 + 1.35i)T + 17iT^{2} \)
19 \( 1 + 3.81iT - 19T^{2} \)
29 \( 1 - 7.78T + 29T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
37 \( 1 + (-1.86 + 1.86i)T - 37iT^{2} \)
41 \( 1 + 4.29iT - 41T^{2} \)
43 \( 1 + (-0.142 - 0.142i)T + 43iT^{2} \)
47 \( 1 + (3.63 + 3.63i)T + 47iT^{2} \)
53 \( 1 + (-3.62 + 3.62i)T - 53iT^{2} \)
59 \( 1 - 3.87T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + (2.77 - 2.77i)T - 67iT^{2} \)
71 \( 1 + 6.15iT - 71T^{2} \)
73 \( 1 + (9.60 + 9.60i)T + 73iT^{2} \)
79 \( 1 - 17.1iT - 79T^{2} \)
83 \( 1 + (-2.00 + 2.00i)T - 83iT^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + (3.36 - 3.36i)T - 97iT^{2} \)
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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

−10.26340127730253163858578224660, −9.497848021073654780207243227764, −9.034850062183637853047406992383, −8.297286406233769809932907889319, −6.98451578681751207516922166262, −6.27765867980796310337516811749, −5.14485061981227063579696008380, −4.36487894061347599872748256129, −2.95327753317412990065238722665, −2.18957233666554753371575864738, 1.16373749488963030491714532372, 2.83300260183442454686979629463, 3.32388609831231282484909768759, 4.28775515490993600613682458446, 6.18430329467611562087526119811, 6.42998629085238166795070310846, 7.56046191486207470846057972415, 8.625128127974482325301123745552, 9.718601416986140490894997999214, 10.23825158273725011481787128656

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