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Properties

Label 2-770-11.5-c1-0-7
Degree $2$
Conductor $770$
Sign $0.998 + 0.0540i$
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 + (0.743 + 0.540i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.284 − 0.874i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.665 − 2.04i)9-s + 0.999·10-s + (3.14 + 1.05i)11-s − 0.919·12-s + (1.15 + 3.54i)13-s + (−0.809 − 0.587i)14-s + (−0.743 + 0.540i)15-s + (0.309 − 0.951i)16-s + (−0.406 + 1.25i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.429 + 0.311i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.115 − 0.356i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (−0.221 − 0.683i)9-s + 0.316·10-s + (0.948 + 0.316i)11-s − 0.265·12-s + (0.319 + 0.982i)13-s + (−0.216 − 0.157i)14-s + (−0.191 + 0.139i)15-s + (0.0772 − 0.237i)16-s + (−0.0987 + 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.998 + 0.0540i$
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: \((2,\ 770,\ (\ :1/2),\ 0.998 + 0.0540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59797 - 0.0432256i\)
\(L(\frac12)\) \(\approx\) \(1.59797 - 0.0432256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.14 - 1.05i)T \)
good3 \( 1 + (-0.743 - 0.540i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (-1.15 - 3.54i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.406 - 1.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.19 - 1.59i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + (-2.38 + 1.73i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.38 - 7.33i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.51 + 4.73i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.62 + 1.90i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.48T + 43T^{2} \)
47 \( 1 + (-7.58 - 5.50i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.10 - 12.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.29 + 1.66i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.446 + 1.37i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + (-2.48 + 7.64i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.07 + 0.782i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.38 + 7.34i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.0881 + 0.271i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.14T + 89T^{2} \)
97 \( 1 + (-4.57 - 14.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.31846096664462745822752351776, −9.355466025838422059480751530371, −8.955010713842334878081784835508, −7.916534932704430454783591303344, −6.89603905861813763870725072918, −5.98580631099300143116876636676, −4.33898626478120665299255404816, −3.86308624689997678398613999300, −2.70518947386529719657560405228, −1.32078709910533543684750428666, 1.03247529145245187578732793201, 2.59980231996983988977444118340, 4.00271241223739133456588652087, 5.12804933310276832158440244677, 5.90351827149758130577189590406, 7.00173674139315160242468936878, 7.948625309708033587344429641826, 8.401441921348913965438951244666, 9.206615708161165115163546352918, 10.12185888069909691132620954682

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