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Properties

Label 2-690-69.68-c1-0-25
Degree $2$
Conductor $690$
Sign $0.986 - 0.165i$
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 + (1.72 + 0.157i)3-s − 4-s + 5-s + (−0.157 + 1.72i)6-s − 4.77i·7-s − i·8-s + (2.95 + 0.544i)9-s + i·10-s + 5.68·11-s + (−1.72 − 0.157i)12-s − 5.86·13-s + 4.77·14-s + (1.72 + 0.157i)15-s + 16-s − 1.96·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.995 + 0.0911i)3-s − 0.5·4-s + 0.447·5-s + (−0.0644 + 0.704i)6-s − 1.80i·7-s − 0.353i·8-s + (0.983 + 0.181i)9-s + 0.316i·10-s + 1.71·11-s + (−0.497 − 0.0455i)12-s − 1.62·13-s + 1.27·14-s + (0.445 + 0.0407i)15-s + 0.250·16-s − 0.476·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.986 - 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20471 + 0.183384i\)
\(L(\frac12)\) \(\approx\) \(2.20471 + 0.183384i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.157i)T \)
5 \( 1 - T \)
23 \( 1 + (1.22 + 4.63i)T \)
good7 \( 1 + 4.77iT - 7T^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
13 \( 1 + 5.86T + 13T^{2} \)
17 \( 1 + 1.96T + 17T^{2} \)
19 \( 1 + 3.08iT - 19T^{2} \)
29 \( 1 - 4.66iT - 29T^{2} \)
31 \( 1 - 7.74T + 31T^{2} \)
37 \( 1 + 3.13iT - 37T^{2} \)
41 \( 1 - 8.70iT - 41T^{2} \)
43 \( 1 - 7.75iT - 43T^{2} \)
47 \( 1 - 8.55iT - 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 - 2.47iT - 59T^{2} \)
61 \( 1 - 2.31iT - 61T^{2} \)
67 \( 1 - 9.67iT - 67T^{2} \)
71 \( 1 - 1.59iT - 71T^{2} \)
73 \( 1 - 3.25T + 73T^{2} \)
79 \( 1 + 6.19iT - 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 7.42T + 89T^{2} \)
97 \( 1 - 6.65iT - 97T^{2} \)
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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

−10.02441376116296700916011580668, −9.653772294082984720192051563160, −8.756579379200153646861952325979, −7.74587071363187635985946755292, −6.95550483325745785604257098667, −6.53647341344546626936342729005, −4.52201515677740788163684503322, −4.36478867454716823249976405272, −2.91492374500158224361663294711, −1.21207209704899208871012570858, 1.83350173762269505275834147506, 2.44469537870079846632513977789, 3.61044855000192947615725115651, 4.78452818467919726508528876111, 5.90726830700692529879329816897, 6.96323760088032607407500532176, 8.237592122465956899933856177964, 8.936052304579827909872610640547, 9.568496916841325994959235350803, 10.01558363590222133583973876229

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