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Properties

Label 2-690-15.14-c2-0-9
Degree $2$
Conductor $690$
Sign $0.240 - 0.970i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (−2.99 + 0.00596i)3-s + 2.00·4-s + (1.21 − 4.85i)5-s + (4.24 − 0.00842i)6-s + 6.90i·7-s − 2.82·8-s + (8.99 − 0.0357i)9-s + (−1.71 + 6.86i)10-s − 1.93i·11-s + (−5.99 + 0.0119i)12-s + 4.38i·13-s − 9.75i·14-s + (−3.60 + 14.5i)15-s + 4.00·16-s + 6.60·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.999 + 0.00198i)3-s + 0.500·4-s + (0.242 − 0.970i)5-s + (0.707 − 0.00140i)6-s + 0.985i·7-s − 0.353·8-s + (0.999 − 0.00397i)9-s + (−0.171 + 0.686i)10-s − 0.175i·11-s + (−0.499 + 0.000993i)12-s + 0.337i·13-s − 0.697i·14-s + (−0.240 + 0.970i)15-s + 0.250·16-s + 0.388·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.240 - 0.970i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.240 - 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6296552668\)
\(L(\frac12)\) \(\approx\) \(0.6296552668\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (2.99 - 0.00596i)T \)
5 \( 1 + (-1.21 + 4.85i)T \)
23 \( 1 + 4.79T \)
good7 \( 1 - 6.90iT - 49T^{2} \)
11 \( 1 + 1.93iT - 121T^{2} \)
13 \( 1 - 4.38iT - 169T^{2} \)
17 \( 1 - 6.60T + 289T^{2} \)
19 \( 1 + 16.9T + 361T^{2} \)
29 \( 1 - 5.45iT - 841T^{2} \)
31 \( 1 - 19.1T + 961T^{2} \)
37 \( 1 + 4.29iT - 1.36e3T^{2} \)
41 \( 1 + 32.7iT - 1.68e3T^{2} \)
43 \( 1 - 29.3iT - 1.84e3T^{2} \)
47 \( 1 + 23.8T + 2.20e3T^{2} \)
53 \( 1 + 28.4T + 2.80e3T^{2} \)
59 \( 1 - 82.3iT - 3.48e3T^{2} \)
61 \( 1 - 64.8T + 3.72e3T^{2} \)
67 \( 1 - 62.3iT - 4.48e3T^{2} \)
71 \( 1 - 130. iT - 5.04e3T^{2} \)
73 \( 1 - 10.5iT - 5.32e3T^{2} \)
79 \( 1 - 72.0T + 6.24e3T^{2} \)
83 \( 1 + 93.0T + 6.88e3T^{2} \)
89 \( 1 - 164. iT - 7.92e3T^{2} \)
97 \( 1 - 69.5iT - 9.40e3T^{2} \)
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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

−10.34618295284187709567712197704, −9.581536858472855518848881595959, −8.796379128705457257134345667825, −8.054602252084329142589889471341, −6.81615499460327416672248086684, −5.92809654994104675580421757632, −5.27549826633091256252535839018, −4.15449750113637826142014153715, −2.27595342161964263357884290359, −1.07219971362631202760995490232, 0.38195515812467144510865333480, 1.84220110726646688633482415399, 3.39425291015923107844004265866, 4.59594824144222750065668731127, 5.91887148058080976102254212202, 6.63067403214621619061947665524, 7.32759547632804044035431264512, 8.152673520370997429034984096580, 9.659965347907918584297743092084, 10.18417880216946577591865150182

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