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Properties

Label 2-680-680.509-c1-0-74
Degree $2$
Conductor $680$
Sign $-0.734 + 0.678i$
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  + (−0.0956 − 1.41i)2-s − 2.33i·3-s + (−1.98 + 0.269i)4-s + (1.97 + 1.04i)5-s + (−3.29 + 0.223i)6-s + 4.43·7-s + (0.570 + 2.77i)8-s − 2.44·9-s + (1.28 − 2.88i)10-s − 2.45·11-s + (0.629 + 4.62i)12-s + 3.83·13-s + (−0.424 − 6.25i)14-s + (2.44 − 4.60i)15-s + (3.85 − 1.06i)16-s + (−3.53 − 2.12i)17-s + ⋯
L(s)  = 1  + (−0.0676 − 0.997i)2-s − 1.34i·3-s + (−0.990 + 0.134i)4-s + (0.883 + 0.468i)5-s + (−1.34 + 0.0910i)6-s + 1.67·7-s + (0.201 + 0.979i)8-s − 0.813·9-s + (0.407 − 0.913i)10-s − 0.741·11-s + (0.181 + 1.33i)12-s + 1.06·13-s + (−0.113 − 1.67i)14-s + (0.630 − 1.19i)15-s + (0.963 − 0.267i)16-s + (−0.856 − 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.734 + 0.678i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653115 - 1.67065i\)
\(L(\frac12)\) \(\approx\) \(0.653115 - 1.67065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0956 + 1.41i)T \)
5 \( 1 + (-1.97 - 1.04i)T \)
17 \( 1 + (3.53 + 2.12i)T \)
good3 \( 1 + 2.33iT - 3T^{2} \)
7 \( 1 - 4.43T + 7T^{2} \)
11 \( 1 + 2.45T + 11T^{2} \)
13 \( 1 - 3.83T + 13T^{2} \)
19 \( 1 + 6.37iT - 19T^{2} \)
23 \( 1 - 6.99T + 23T^{2} \)
29 \( 1 + 9.25T + 29T^{2} \)
31 \( 1 - 7.14iT - 31T^{2} \)
37 \( 1 - 1.05iT - 37T^{2} \)
41 \( 1 - 5.40iT - 41T^{2} \)
43 \( 1 + 1.71T + 43T^{2} \)
47 \( 1 + 1.62iT - 47T^{2} \)
53 \( 1 + 3.10T + 53T^{2} \)
59 \( 1 + 0.215iT - 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 - 9.28T + 67T^{2} \)
71 \( 1 - 6.57iT - 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 3.29iT - 79T^{2} \)
83 \( 1 + 2.67T + 83T^{2} \)
89 \( 1 + 4.86T + 89T^{2} \)
97 \( 1 - 5.84T + 97T^{2} \)
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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

−10.55184493666887608385006626513, −9.151095536690400152971747747623, −8.563089892119924019331848450748, −7.58202137458675678054716294064, −6.78711819174719174428021583622, −5.45411809683365791934481718621, −4.73243462126899271156341450700, −2.91783867221819109424172105970, −2.00029936534978262581605493765, −1.18524304314792319684110354990, 1.66055584069045433661897540456, 3.83083781007505408217599915487, 4.63201843188864540995070045969, 5.38249963967475433763120341202, 5.95552744245029202233075245261, 7.51173390295616691826002261930, 8.428126736587873328543237566046, 8.939708107998780886980373882041, 9.798908410930405092017488894920, 10.70523507108431089321658868143

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