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Properties

Label 2-770-11.9-c1-0-7
Degree $2$
Conductor $770$
Sign $0.823 - 0.567i$
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  + (0.309 − 0.951i)2-s + (−0.295 + 0.214i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.113 + 0.347i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.885 + 2.72i)9-s − 0.999·10-s + (−2.13 + 2.53i)11-s + 0.365·12-s + (−0.679 + 2.09i)13-s + (0.809 − 0.587i)14-s + (0.295 + 0.214i)15-s + (0.309 + 0.951i)16-s + (1.93 + 5.95i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.170 + 0.124i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.0461 + 0.142i)6-s + (0.305 + 0.222i)7-s + (−0.286 + 0.207i)8-s + (−0.295 + 0.908i)9-s − 0.316·10-s + (−0.643 + 0.765i)11-s + 0.105·12-s + (−0.188 + 0.580i)13-s + (0.216 − 0.157i)14-s + (0.0764 + 0.0555i)15-s + (0.0772 + 0.237i)16-s + (0.469 + 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17566 + 0.366164i\)
\(L(\frac12)\) \(\approx\) \(1.17566 + 0.366164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (2.13 - 2.53i)T \)
good3 \( 1 + (0.295 - 0.214i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (0.679 - 2.09i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.93 - 5.95i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.86 + 3.53i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.452T + 23T^{2} \)
29 \( 1 + (-1.54 - 1.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.86 - 5.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.86 + 2.08i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.0167 + 0.0121i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.71T + 43T^{2} \)
47 \( 1 + (6.98 - 5.07i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.48 + 7.64i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.2 - 7.48i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.86 - 5.72i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 8.19T + 67T^{2} \)
71 \( 1 + (-4.87 - 14.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.3 + 7.51i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.32 + 10.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.38 - 10.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.56T + 89T^{2} \)
97 \( 1 + (1.30 - 4.01i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.48621883021334664012138879112, −9.751353502460292565487641402516, −8.741806536559387829282991628622, −8.000412802318815833837733645573, −6.99296222349542389614057573304, −5.50065715978672240464687758939, −5.03074606580143655003058286328, −4.04354954028308965770273850940, −2.67193493763276108948165121550, −1.56094933278138180448642035520, 0.61898231254880220344717827039, 2.89868773776803641688834321063, 3.69331688653075553268656802794, 5.16189610930018072237859498681, 5.73675344888975407683699683342, 6.78313046134929094259333677622, 7.60930893032240898221756472250, 8.237187441093000506380066291967, 9.393012590843062842537671976965, 10.09353535302599572273740927235

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