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Properties

Label 2-680-136.101-c1-0-42
Degree $2$
Conductor $680$
Sign $0.0179 - 0.999i$
Analytic cond. $5.42982$
Root an. cond. $2.33019$
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  + (0.458 + 1.33i)2-s + 2.54·3-s + (−1.57 + 1.22i)4-s + 5-s + (1.16 + 3.40i)6-s + 0.654i·7-s + (−2.36 − 1.54i)8-s + 3.48·9-s + (0.458 + 1.33i)10-s + 2.31·11-s + (−4.02 + 3.12i)12-s − 0.495i·13-s + (−0.875 + 0.300i)14-s + 2.54·15-s + (0.987 − 3.87i)16-s + (2.19 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.324 + 0.945i)2-s + 1.47·3-s + (−0.789 + 0.613i)4-s + 0.447·5-s + (0.477 + 1.39i)6-s + 0.247i·7-s + (−0.836 − 0.547i)8-s + 1.16·9-s + (0.145 + 0.423i)10-s + 0.698·11-s + (−1.16 + 0.902i)12-s − 0.137i·13-s + (−0.233 + 0.0802i)14-s + 0.657·15-s + (0.246 − 0.969i)16-s + (0.532 + 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0179 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0179 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.0179 - 0.999i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.0179 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00862 + 1.97292i\)
\(L(\frac12)\) \(\approx\) \(2.00862 + 1.97292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.458 - 1.33i)T \)
5 \( 1 - T \)
17 \( 1 + (-2.19 - 3.48i)T \)
good3 \( 1 - 2.54T + 3T^{2} \)
7 \( 1 - 0.654iT - 7T^{2} \)
11 \( 1 - 2.31T + 11T^{2} \)
13 \( 1 + 0.495iT - 13T^{2} \)
19 \( 1 - 2.57iT - 19T^{2} \)
23 \( 1 - 0.188iT - 23T^{2} \)
29 \( 1 + 6.85T + 29T^{2} \)
31 \( 1 + 3.96iT - 31T^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 + 3.21iT - 41T^{2} \)
43 \( 1 + 3.55iT - 43T^{2} \)
47 \( 1 + 3.43T + 47T^{2} \)
53 \( 1 + 8.47iT - 53T^{2} \)
59 \( 1 + 0.432iT - 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 - 9.72iT - 71T^{2} \)
73 \( 1 + 3.67iT - 73T^{2} \)
79 \( 1 + 4.08iT - 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 - 8.98T + 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−10.29268749871852309889842390104, −9.409935783428150124925301536975, −8.873875414286839551134021560803, −8.078702549978755223377043455138, −7.38283163948510623945711219957, −6.28175370348546032170530714946, −5.41792303971835802701138018762, −4.00158226416801716400224160619, −3.36441196241890042390326709240, −1.95455615244472757676455122963, 1.41568696759676052838864912769, 2.56584530779796735192174185892, 3.39833988279422328714206433951, 4.34477121675495634825554821297, 5.49755427971003386772643575159, 6.83363834626662337865593734799, 7.901506269964771422417190782014, 9.006676529383622512198308765435, 9.298639948433322934746871420929, 10.11616114507044132956930521160

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