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Properties

Label 2-1275-17.16-c1-0-46
Degree $2$
Conductor $1275$
Sign $-0.970 - 0.242i$
Analytic cond. $10.1809$
Root an. cond. $3.19075$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + i·3-s − 4-s − i·6-s − 4i·7-s + 3·8-s − 9-s − 4i·11-s − i·12-s − 2·13-s + 4i·14-s − 16-s + (−1 + 4i)17-s + 18-s − 4·19-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s − 0.5·4-s − 0.408i·6-s − 1.51i·7-s + 1.06·8-s − 0.333·9-s − 1.20i·11-s − 0.288i·12-s − 0.554·13-s + 1.06i·14-s − 0.250·16-s + (−0.242 + 0.970i)17-s + 0.235·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(10.1809\)
Root analytic conductor: \(3.19075\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1275,\ (\ :1/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
17 \( 1 + (1 - 4i)T \)
good2 \( 1 + T + 2T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−9.240865333537153393247348535350, −8.438139548596450701533956956059, −7.940202840837359735727103398567, −6.89659493237785739410687667303, −5.95680565304296858032361522712, −4.59505012688846537584779410123, −4.22871218028190411416718814833, −3.12051466122388402604787547890, −1.26849342818630215934957186729, 0, 1.82748590701356091986726448753, 2.56955017673526858066482694457, 4.23608399366112441038375086731, 5.14373402617204530909179652607, 5.93571938850175793564176660244, 7.19846785058994833652516209944, 7.63876662426734180254602478407, 8.705702605649184997899163573038, 9.208760012218860059160387556555

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