Properties

Label 2-690-69.68-c1-0-21
Degree $2$
Conductor $690$
Sign $0.962 + 0.272i$
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 + (1.05 + 1.37i)3-s − 4-s + 5-s + (1.37 − 1.05i)6-s − 1.91i·7-s + i·8-s + (−0.772 + 2.89i)9-s i·10-s + 2.07·11-s + (−1.05 − 1.37i)12-s + 1.28·13-s − 1.91·14-s + (1.05 + 1.37i)15-s + 16-s + 4.04·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.609 + 0.792i)3-s − 0.5·4-s + 0.447·5-s + (0.560 − 0.430i)6-s − 0.722i·7-s + 0.353i·8-s + (−0.257 + 0.966i)9-s − 0.316i·10-s + 0.625·11-s + (−0.304 − 0.396i)12-s + 0.357·13-s − 0.510·14-s + (0.272 + 0.354i)15-s + 0.250·16-s + 0.981·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.962 + 0.272i$
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: \((2,\ 690,\ (\ :1/2),\ 0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97298 - 0.274231i\)
\(L(\frac12)\) \(\approx\) \(1.97298 - 0.274231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 - 1.37i)T \)
5 \( 1 - T \)
23 \( 1 + (-2.86 - 3.84i)T \)
good7 \( 1 + 1.91iT - 7T^{2} \)
11 \( 1 - 2.07T + 11T^{2} \)
13 \( 1 - 1.28T + 13T^{2} \)
17 \( 1 - 4.04T + 17T^{2} \)
19 \( 1 + 3.79iT - 19T^{2} \)
29 \( 1 - 2.78iT - 29T^{2} \)
31 \( 1 - 1.87T + 31T^{2} \)
37 \( 1 + 2.18iT - 37T^{2} \)
41 \( 1 + 0.590iT - 41T^{2} \)
43 \( 1 - 0.332iT - 43T^{2} \)
47 \( 1 - 4.11iT - 47T^{2} \)
53 \( 1 + 3.47T + 53T^{2} \)
59 \( 1 - 2.07iT - 59T^{2} \)
61 \( 1 - 3.23iT - 61T^{2} \)
67 \( 1 + 6.73iT - 67T^{2} \)
71 \( 1 + 9.09iT - 71T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 + 3.38iT - 79T^{2} \)
83 \( 1 + 5.68T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−10.47213270115799886618682024229, −9.531654208593007426298489702947, −9.110188413935564797248596881997, −8.057146197320199252835256620802, −7.05288458929578402738562472064, −5.65923050776381015296540870159, −4.66718543369286319136492505100, −3.72647006796067120408727223109, −2.86952549065850153196892441994, −1.37041968194304937593246222976, 1.32236868441983736874280895923, 2.72928049917321481118621344933, 3.92867193340263971621352900633, 5.42914527873735709747683767694, 6.18241087290947600044085267262, 6.91955654748199607783066393487, 7.989437748042924567712357137936, 8.600972266968155186514125021847, 9.374156411943099002651397352917, 10.18823436143214071930214559484

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