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Properties

Label 2-690-15.2-c1-0-8
Degree $2$
Conductor $690$
Sign $0.938 - 0.345i$
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 + (−1.16 − 1.28i)3-s − 1.00i·4-s + (−1.31 − 1.80i)5-s + (1.73 + 0.0807i)6-s + (1.43 + 1.43i)7-s + (0.707 + 0.707i)8-s + (−0.279 + 2.98i)9-s + (2.20 + 0.344i)10-s + 3.03i·11-s + (−1.28 + 1.16i)12-s + (−1.30 + 1.30i)13-s − 2.03·14-s + (−0.774 + 3.79i)15-s − 1.00·16-s + (0.0287 − 0.0287i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.673 − 0.739i)3-s − 0.500i·4-s + (−0.589 − 0.807i)5-s + (0.706 + 0.0329i)6-s + (0.542 + 0.542i)7-s + (0.250 + 0.250i)8-s + (−0.0931 + 0.995i)9-s + (0.698 + 0.108i)10-s + 0.915i·11-s + (−0.369 + 0.336i)12-s + (−0.361 + 0.361i)13-s − 0.542·14-s + (−0.199 + 0.979i)15-s − 0.250·16-s + (0.00696 − 0.00696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.938 - 0.345i$
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: \((2,\ 690,\ (\ :1/2),\ 0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781799 + 0.139132i\)
\(L(\frac12)\) \(\approx\) \(0.781799 + 0.139132i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 + 1.28i)T \)
5 \( 1 + (1.31 + 1.80i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-1.43 - 1.43i)T + 7iT^{2} \)
11 \( 1 - 3.03iT - 11T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - 13iT^{2} \)
17 \( 1 + (-0.0287 + 0.0287i)T - 17iT^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + (0.399 + 0.399i)T + 37iT^{2} \)
41 \( 1 - 9.45iT - 41T^{2} \)
43 \( 1 + (0.108 - 0.108i)T - 43iT^{2} \)
47 \( 1 + (-0.341 + 0.341i)T - 47iT^{2} \)
53 \( 1 + (-0.120 - 0.120i)T + 53iT^{2} \)
59 \( 1 - 2.10T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + (6.68 + 6.68i)T + 67iT^{2} \)
71 \( 1 - 5.69iT - 71T^{2} \)
73 \( 1 + (-6.43 + 6.43i)T - 73iT^{2} \)
79 \( 1 - 5.14iT - 79T^{2} \)
83 \( 1 + (3.07 + 3.07i)T + 83iT^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-7.38 - 7.38i)T + 97iT^{2} \)
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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

−10.47395787816822725727016800763, −9.552868883217658222445561847199, −8.456198180671581680145057357776, −8.003819598905310857568754382716, −7.03892901350444410011476769155, −6.26072317170955490071735984235, −4.96416689091990257740150688034, −4.65518940982763082216011633923, −2.31942845918856864561531555742, −1.00546426208648743728414424696, 0.73084159656145798254700218379, 2.84193085485005562189891967114, 3.79896971789361312011166003354, 4.69605099555453431522946047925, 5.99173365096781500360057317348, 6.94677673090637124433091362481, 7.986592354799843364789856399580, 8.684836167935899837128563181619, 10.09781349503536349529319537105, 10.32548715031889191448538338568

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