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Properties

Label 2-770-385.164-c1-0-28
Degree $2$
Conductor $770$
Sign $0.826 + 0.562i$
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  + (−0.5 − 0.866i)2-s + (−1.65 + 2.86i)3-s + (−0.499 + 0.866i)4-s + (−0.232 + 2.22i)5-s + 3.30·6-s + (2.27 − 1.35i)7-s + 0.999·8-s + (−3.95 − 6.85i)9-s + (2.04 − 0.910i)10-s + (0.366 − 3.29i)11-s + (−1.65 − 2.86i)12-s − 4.25i·13-s + (−2.30 − 1.29i)14-s + (−5.97 − 4.33i)15-s + (−0.5 − 0.866i)16-s + (2.87 + 1.66i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.953 + 1.65i)3-s + (−0.249 + 0.433i)4-s + (−0.103 + 0.994i)5-s + 1.34·6-s + (0.858 − 0.512i)7-s + 0.353·8-s + (−1.31 − 2.28i)9-s + (0.645 − 0.288i)10-s + (0.110 − 0.993i)11-s + (−0.476 − 0.825i)12-s − 1.18i·13-s + (−0.617 − 0.344i)14-s + (−1.54 − 1.12i)15-s + (−0.125 − 0.216i)16-s + (0.697 + 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.826 + 0.562i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704784 - 0.216994i\)
\(L(\frac12)\) \(\approx\) \(0.704784 - 0.216994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.232 - 2.22i)T \)
7 \( 1 + (-2.27 + 1.35i)T \)
11 \( 1 + (-0.366 + 3.29i)T \)
good3 \( 1 + (1.65 - 2.86i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 4.25iT - 13T^{2} \)
17 \( 1 + (-2.87 - 1.66i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.63 + 4.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.51 + 1.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.72iT - 29T^{2} \)
31 \( 1 + (-2.18 - 1.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.41 - 1.39i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.35T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 + (1.46 + 2.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.72 - 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.45 + 3.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.02 + 5.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.8 + 6.85i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.0378T + 71T^{2} \)
73 \( 1 + (-11.9 - 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.23 - 2.44i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.42iT - 83T^{2} \)
89 \( 1 + (-10.1 + 5.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.07T + 97T^{2} \)
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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

−10.44539200220381017194161451082, −9.855586183239630696380773447539, −8.733428848301745204650978103389, −7.920479273658627706606437778682, −6.54645755677102868351068086526, −5.59842354488729582734399700125, −4.65997458003216535110237311635, −3.70867416162111633173880565040, −2.94288641577295486303934216460, −0.53134738566374121485499395438, 1.32008560727071495509500759053, 1.91928863910042089088940424974, 4.59387619112248579641271727315, 5.27070174972660937270347220011, 6.07940180977560113305351755491, 7.05287514955344328144542726613, 7.66760047341891735059911706623, 8.432995172538411802655736583715, 9.209223667192334358453709312868, 10.49086711481158440954545309069

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