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Properties

Label 2-770-55.32-c1-0-21
Degree $2$
Conductor $770$
Sign $0.0489 + 0.998i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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 + (0.129 − 0.129i)3-s − 1.00i·4-s + (−1.76 + 1.36i)5-s + 0.182i·6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 2.96i·9-s + (0.282 − 2.21i)10-s + (−1.31 − 3.04i)11-s + (−0.129 − 0.129i)12-s + (−2.63 − 2.63i)13-s − 1.00i·14-s + (−0.0516 + 0.404i)15-s − 1.00·16-s + (3.32 − 3.32i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.0745 − 0.0745i)3-s − 0.500i·4-s + (−0.790 + 0.611i)5-s + 0.0745i·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.988i·9-s + (0.0894 − 0.701i)10-s + (−0.396 − 0.918i)11-s + (−0.0372 − 0.0372i)12-s + (−0.731 − 0.731i)13-s − 0.267i·14-s + (−0.0133 + 0.104i)15-s − 0.250·16-s + (0.806 − 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0489 + 0.998i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.0489 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267474 - 0.254674i\)
\(L(\frac12)\) \(\approx\) \(0.267474 - 0.254674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.76 - 1.36i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (1.31 + 3.04i)T \)
good3 \( 1 + (-0.129 + 0.129i)T - 3iT^{2} \)
13 \( 1 + (2.63 + 2.63i)T + 13iT^{2} \)
17 \( 1 + (-3.32 + 3.32i)T - 17iT^{2} \)
19 \( 1 + 2.52T + 19T^{2} \)
23 \( 1 + (1.90 - 1.90i)T - 23iT^{2} \)
29 \( 1 + 0.802T + 29T^{2} \)
31 \( 1 - 9.02T + 31T^{2} \)
37 \( 1 + (7.49 + 7.49i)T + 37iT^{2} \)
41 \( 1 + 5.93iT - 41T^{2} \)
43 \( 1 + (7.13 + 7.13i)T + 43iT^{2} \)
47 \( 1 + (-1.82 - 1.82i)T + 47iT^{2} \)
53 \( 1 + (0.115 - 0.115i)T - 53iT^{2} \)
59 \( 1 - 2.90iT - 59T^{2} \)
61 \( 1 + 6.91iT - 61T^{2} \)
67 \( 1 + (11.2 + 11.2i)T + 67iT^{2} \)
71 \( 1 + 0.437T + 71T^{2} \)
73 \( 1 + (7.08 + 7.08i)T + 73iT^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 + (-5.44 - 5.44i)T + 83iT^{2} \)
89 \( 1 - 2.70iT - 89T^{2} \)
97 \( 1 + (-1.36 - 1.36i)T + 97iT^{2} \)
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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

−10.40503405260442174864166369504, −9.111560145521547555418590062082, −8.107920280101045256473858718722, −7.72229003321674861790112509108, −6.84758732020778021833187913754, −5.71876165432579554478814715584, −4.92396649839533494666825620662, −3.41969538069851271029019239642, −2.41190990704330180598984150235, −0.22462963415323087525451669529, 1.40928507067386327412876955071, 2.98218507509540925957396012063, 4.08591444203402057951014640778, 4.75886823494174127834651268827, 6.36823076410862048513444516111, 7.22640977879225646879025900073, 8.158394179877249416820388537655, 8.807583345186104457859532562614, 9.933082042941797725383591521292, 10.12560873590320937274861735406

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