Properties

Label 2-690-15.2-c1-0-35
Degree $2$
Conductor $690$
Sign $-0.432 + 0.901i$
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 + (0.396 + 1.68i)3-s − 1.00i·4-s + (−2.10 + 0.743i)5-s + (1.47 + 0.912i)6-s + (−2.17 − 2.17i)7-s + (−0.707 − 0.707i)8-s + (−2.68 + 1.33i)9-s + (−0.965 + 2.01i)10-s − 5.22i·11-s + (1.68 − 0.396i)12-s + (2.60 − 2.60i)13-s − 3.06·14-s + (−2.08 − 3.26i)15-s − 1.00·16-s + (−0.444 + 0.444i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.228 + 0.973i)3-s − 0.500i·4-s + (−0.943 + 0.332i)5-s + (0.601 + 0.372i)6-s + (−0.820 − 0.820i)7-s + (−0.250 − 0.250i)8-s + (−0.895 + 0.445i)9-s + (−0.305 + 0.637i)10-s − 1.57i·11-s + (0.486 − 0.114i)12-s + (0.721 − 0.721i)13-s − 0.820·14-s + (−0.539 − 0.842i)15-s − 0.250·16-s + (−0.107 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.432 + 0.901i$
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: \((2,\ 690,\ (\ :1/2),\ -0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547908 - 0.870853i\)
\(L(\frac12)\) \(\approx\) \(0.547908 - 0.870853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.396 - 1.68i)T \)
5 \( 1 + (2.10 - 0.743i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (2.17 + 2.17i)T + 7iT^{2} \)
11 \( 1 + 5.22iT - 11T^{2} \)
13 \( 1 + (-2.60 + 2.60i)T - 13iT^{2} \)
17 \( 1 + (0.444 - 0.444i)T - 17iT^{2} \)
19 \( 1 + 2.97iT - 19T^{2} \)
29 \( 1 + 1.70T + 29T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 + (2.75 + 2.75i)T + 37iT^{2} \)
41 \( 1 - 6.96iT - 41T^{2} \)
43 \( 1 + (-7.32 + 7.32i)T - 43iT^{2} \)
47 \( 1 + (6.05 - 6.05i)T - 47iT^{2} \)
53 \( 1 + (0.302 + 0.302i)T + 53iT^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 + (-3.64 - 3.64i)T + 67iT^{2} \)
71 \( 1 + 4.61iT - 71T^{2} \)
73 \( 1 + (-0.698 + 0.698i)T - 73iT^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 + (-9.38 - 9.38i)T + 83iT^{2} \)
89 \( 1 + 3.53T + 89T^{2} \)
97 \( 1 + (13.1 + 13.1i)T + 97iT^{2} \)
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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

−10.68110881028937419416543240629, −9.439571693152915412722753491357, −8.629626238913302429111747854540, −7.67828089841547346075281907092, −6.43978053821306171765164507774, −5.51832460561964338529815948257, −4.27492472759672635308484963066, −3.44643417532509307319776871352, −3.07971083454828396193711019463, −0.43720145741725400777229813823, 1.89947239363884291751664265648, 3.24179639522399669516269657078, 4.26271738502075832010132571290, 5.48047220001929290923132515403, 6.49965834074480453431170617592, 7.15115211076339217950726158183, 7.946091243532485352602845185921, 8.858715323253626423443493831408, 9.498537627904475979776949465905, 11.08297616902649592983368372628

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