Properties

Label 2-690-15.2-c1-0-43
Degree $2$
Conductor $690$
Sign $-0.858 - 0.513i$
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 + (−1.00 − 1.40i)3-s − 1.00i·4-s + (0.318 − 2.21i)5-s + (−1.70 − 0.283i)6-s + (−2.49 − 2.49i)7-s + (−0.707 − 0.707i)8-s + (−0.968 + 2.83i)9-s + (−1.34 − 1.79i)10-s − 0.391i·11-s + (−1.40 + 1.00i)12-s + (−0.323 + 0.323i)13-s − 3.53·14-s + (−3.43 + 1.78i)15-s − 1.00·16-s + (3.34 − 3.34i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.581 − 0.813i)3-s − 0.500i·4-s + (0.142 − 0.989i)5-s + (−0.697 − 0.115i)6-s + (−0.943 − 0.943i)7-s + (−0.250 − 0.250i)8-s + (−0.322 + 0.946i)9-s + (−0.423 − 0.566i)10-s − 0.117i·11-s + (−0.406 + 0.290i)12-s + (−0.0897 + 0.0897i)13-s − 0.943·14-s + (−0.887 + 0.460i)15-s − 0.250·16-s + (0.810 − 0.810i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.858 - 0.513i$
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: \((2,\ 690,\ (\ :1/2),\ -0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284087 + 1.02868i\)
\(L(\frac12)\) \(\approx\) \(0.284087 + 1.02868i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.00 + 1.40i)T \)
5 \( 1 + (-0.318 + 2.21i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (2.49 + 2.49i)T + 7iT^{2} \)
11 \( 1 + 0.391iT - 11T^{2} \)
13 \( 1 + (0.323 - 0.323i)T - 13iT^{2} \)
17 \( 1 + (-3.34 + 3.34i)T - 17iT^{2} \)
19 \( 1 - 7.09iT - 19T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 3.88T + 31T^{2} \)
37 \( 1 + (0.830 + 0.830i)T + 37iT^{2} \)
41 \( 1 + 9.20iT - 41T^{2} \)
43 \( 1 + (-4.32 + 4.32i)T - 43iT^{2} \)
47 \( 1 + (7.82 - 7.82i)T - 47iT^{2} \)
53 \( 1 + (-1.05 - 1.05i)T + 53iT^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 1.98T + 61T^{2} \)
67 \( 1 + (6.39 + 6.39i)T + 67iT^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 + (-5.64 + 5.64i)T - 73iT^{2} \)
79 \( 1 - 8.35iT - 79T^{2} \)
83 \( 1 + (4.16 + 4.16i)T + 83iT^{2} \)
89 \( 1 + 1.17T + 89T^{2} \)
97 \( 1 + (3.27 + 3.27i)T + 97iT^{2} \)
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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

−10.14010180710666434009551408344, −9.327127988476740922955143441004, −8.071807239693966256657426949798, −7.23663473257827396635629882804, −6.18488072769961792448577799188, −5.48393439410119194220059516407, −4.42297175408339695829337860263, −3.29304172290253754086028942854, −1.68878373871504059904974293699, −0.51014412290213530603835328972, 2.75354830206331381918634073320, 3.45488638138781944138802745484, 4.72011337771006816221432715465, 5.75505734944757067577910109870, 6.32310157462797574699079235017, 7.09599137435012017778891857573, 8.437038548837454620004491217952, 9.443595188484542514664168301771, 10.04114559840565637023939936003, 11.01759263239334827011498218203

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