Properties

Label 2-770-385.152-c1-0-46
Degree $2$
Conductor $770$
Sign $-0.892 - 0.451i$
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  + (0.0523 − 0.998i)2-s + (0.0281 + 0.0732i)3-s + (−0.994 − 0.104i)4-s + (0.0572 − 2.23i)5-s + (0.0745 − 0.0242i)6-s + (−2.64 + 0.147i)7-s + (−0.156 + 0.987i)8-s + (2.22 − 2.00i)9-s + (−2.22 − 0.174i)10-s + (−1.09 − 3.12i)11-s + (−0.0202 − 0.0757i)12-s + (−3.57 + 1.82i)13-s + (0.00867 + 2.64i)14-s + (0.165 − 0.0586i)15-s + (0.978 + 0.207i)16-s + (0.0428 + 0.817i)17-s + ⋯
L(s)  = 1  + (0.0370 − 0.706i)2-s + (0.0162 + 0.0422i)3-s + (−0.497 − 0.0522i)4-s + (0.0256 − 0.999i)5-s + (0.0304 − 0.00989i)6-s + (−0.998 + 0.0556i)7-s + (−0.0553 + 0.349i)8-s + (0.741 − 0.667i)9-s + (−0.704 − 0.0550i)10-s + (−0.331 − 0.943i)11-s + (−0.00585 − 0.0218i)12-s + (−0.992 + 0.505i)13-s + (0.00231 + 0.707i)14-s + (0.0426 − 0.0151i)15-s + (0.244 + 0.0519i)16-s + (0.0103 + 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145396 + 0.608824i\)
\(L(\frac12)\) \(\approx\) \(0.145396 + 0.608824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0523 + 0.998i)T \)
5 \( 1 + (-0.0572 + 2.23i)T \)
7 \( 1 + (2.64 - 0.147i)T \)
11 \( 1 + (1.09 + 3.12i)T \)
good3 \( 1 + (-0.0281 - 0.0732i)T + (-2.22 + 2.00i)T^{2} \)
13 \( 1 + (3.57 - 1.82i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.0428 - 0.817i)T + (-16.9 + 1.77i)T^{2} \)
19 \( 1 + (-0.526 - 5.01i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (0.947 + 3.53i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.74 - 2.40i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.229 + 1.08i)T + (-28.3 + 12.6i)T^{2} \)
37 \( 1 + (8.52 + 3.27i)T + (27.4 + 24.7i)T^{2} \)
41 \( 1 + (5.24 + 7.21i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.69 - 6.69i)T - 43iT^{2} \)
47 \( 1 + (-3.49 - 2.83i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (-5.04 + 7.76i)T + (-21.5 - 48.4i)T^{2} \)
59 \( 1 + (0.699 - 6.65i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-1.46 + 6.88i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (1.27 - 4.76i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.12 + 9.61i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.15 - 0.935i)T + (15.1 - 71.4i)T^{2} \)
79 \( 1 + (3.10 - 2.79i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (-1.53 + 3.01i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-5.03 + 8.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.579 - 1.13i)T + (-57.0 + 78.4i)T^{2} \)
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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

−9.915766357951867060038040966287, −9.108793890738109908334211961951, −8.457720147724949937684202644760, −7.25208992523510001019253939033, −6.14889391502467730897527008718, −5.21710871366684657997879352738, −4.10150659311855015825074978778, −3.31461693203186976093012771831, −1.80853136263953169977917232909, −0.29849905442693691411188631635, 2.29088019175577377103229905069, 3.38477372388742138155861351251, 4.65665588873352729757458324128, 5.53237617084072889550587000811, 6.89886338907689914296493191721, 7.05975996997233105797605975092, 7.88196377177073981104297396838, 9.235603169188405063861970696238, 10.13587975760368245640363343346, 10.28218803388874865156707672226

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