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Properties

Label 2-770-11.5-c1-0-8
Degree $2$
Conductor $770$
Sign $-0.690 - 0.722i$
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 + (2.05 + 1.49i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.784 + 2.41i)6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (1.06 + 3.27i)9-s − 0.999·10-s + (1.57 + 2.91i)11-s − 2.53·12-s + (−1.27 − 3.91i)13-s + (0.809 + 0.587i)14-s + (−2.05 + 1.49i)15-s + (0.309 − 0.951i)16-s + (−2.11 + 6.51i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.18 + 0.861i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.320 + 0.985i)6-s + (0.305 − 0.222i)7-s + (−0.286 − 0.207i)8-s + (0.355 + 1.09i)9-s − 0.316·10-s + (0.474 + 0.880i)11-s − 0.733·12-s + (−0.352 − 1.08i)13-s + (0.216 + 0.157i)14-s + (−0.530 + 0.385i)15-s + (0.0772 − 0.237i)16-s + (−0.513 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.690 - 0.722i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.690 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934623 + 2.18635i\)
\(L(\frac12)\) \(\approx\) \(0.934623 + 2.18635i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 2.91i)T \)
good3 \( 1 + (-2.05 - 1.49i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (1.27 + 3.91i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.11 - 6.51i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.35 - 3.89i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 + (4.44 - 3.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.29 + 4.57i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.35 + 1.70i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (-1.21 - 0.881i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.43 + 10.5i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.00 + 4.36i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.705 - 2.16i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (-0.0188 + 0.0579i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.16 + 2.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.31 + 4.03i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.16 + 9.74i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.54T + 89T^{2} \)
97 \( 1 + (0.213 + 0.657i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.21964483374188084070902003551, −9.783837066155273513942246778697, −8.857665904672378070259785565684, −7.890163017412300903094390971109, −7.57253120987879602966609150650, −6.24364474762782452622018700530, −5.17915943588331624126264056028, −3.97388203303323126201509480746, −3.59785323084635077640562041518, −2.15107040822125609276185906769, 1.06145668071880319308554622809, 2.27902543856462347780028926469, 3.11164497921485364032731539440, 4.30347628576445381861456950155, 5.34458388377896704502489017226, 6.72389932724991184134798610240, 7.52622984215369337970437178150, 8.462858250833557170922230408981, 9.215974446801449094896763215627, 9.563590951674839762270732678906

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