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Properties

Label 2-770-35.13-c1-0-10
Degree $2$
Conductor $770$
Sign $0.0380 - 0.999i$
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 + (−1.65 + 1.65i)3-s − 1.00i·4-s + (2.15 + 0.602i)5-s + 2.33i·6-s + (0.00754 + 2.64i)7-s + (−0.707 − 0.707i)8-s − 2.45i·9-s + (1.94 − 1.09i)10-s − 11-s + (1.65 + 1.65i)12-s + (0.174 − 0.174i)13-s + (1.87 + 1.86i)14-s + (−4.55 + 2.56i)15-s − 1.00·16-s + (5.60 + 5.60i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.953 + 0.953i)3-s − 0.500i·4-s + (0.963 + 0.269i)5-s + 0.953i·6-s + (0.00285 + 0.999i)7-s + (−0.250 − 0.250i)8-s − 0.817i·9-s + (0.616 − 0.346i)10-s − 0.301·11-s + (0.476 + 0.476i)12-s + (0.0483 − 0.0483i)13-s + (0.501 + 0.498i)14-s + (−1.17 + 0.661i)15-s − 0.250·16-s + (1.35 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0380 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0380 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0380 - 0.999i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.0380 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04805 + 1.00887i\)
\(L(\frac12)\) \(\approx\) \(1.04805 + 1.00887i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 0.602i)T \)
7 \( 1 + (-0.00754 - 2.64i)T \)
11 \( 1 + T \)
good3 \( 1 + (1.65 - 1.65i)T - 3iT^{2} \)
13 \( 1 + (-0.174 + 0.174i)T - 13iT^{2} \)
17 \( 1 + (-5.60 - 5.60i)T + 17iT^{2} \)
19 \( 1 + 5.84T + 19T^{2} \)
23 \( 1 + (2.78 + 2.78i)T + 23iT^{2} \)
29 \( 1 + 1.55iT - 29T^{2} \)
31 \( 1 - 5.97iT - 31T^{2} \)
37 \( 1 + (3.47 - 3.47i)T - 37iT^{2} \)
41 \( 1 - 5.31iT - 41T^{2} \)
43 \( 1 + (-2.57 - 2.57i)T + 43iT^{2} \)
47 \( 1 + (-3.10 - 3.10i)T + 47iT^{2} \)
53 \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \)
59 \( 1 + 8.26T + 59T^{2} \)
61 \( 1 - 5.87iT - 61T^{2} \)
67 \( 1 + (-6.82 + 6.82i)T - 67iT^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 + (0.699 - 0.699i)T - 73iT^{2} \)
79 \( 1 + 7.29iT - 79T^{2} \)
83 \( 1 + (-8.97 + 8.97i)T - 83iT^{2} \)
89 \( 1 + 4.37T + 89T^{2} \)
97 \( 1 + (-0.622 - 0.622i)T + 97iT^{2} \)
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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

−10.43079254483451169807136783156, −10.14556596985530993168410203707, −9.158808362583960110806124381236, −8.174011198762962073895403151659, −6.36695739916466987831669611678, −5.92537022847008536985649988752, −5.22538629314189941672704369694, −4.29859460788569711283969451235, −3.02698442912320608045601688676, −1.81619579519898590392732539122, 0.70082936314876791544827916493, 2.14688720445124496871011238066, 3.79990760811743727312490022571, 5.11669592604043375139233277181, 5.68350165426842855520953458800, 6.58013988737211232037752588320, 7.22614605118320894995385984578, 8.005837415013563077473151300707, 9.301851729601456274819624726795, 10.21886474156821163698139804995

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