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Properties

Label 2-680-136.101-c1-0-39
Degree $2$
Conductor $680$
Sign $0.735 - 0.677i$
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  + (−1.27 + 0.619i)2-s + 3.03·3-s + (1.23 − 1.57i)4-s + 5-s + (−3.85 + 1.88i)6-s + 1.14i·7-s + (−0.588 + 2.76i)8-s + 6.20·9-s + (−1.27 + 0.619i)10-s + 1.02·11-s + (3.73 − 4.78i)12-s + 5.48i·13-s + (−0.710 − 1.45i)14-s + 3.03·15-s + (−0.967 − 3.88i)16-s + (−3.36 − 2.38i)17-s + ⋯
L(s)  = 1  + (−0.898 + 0.438i)2-s + 1.75·3-s + (0.615 − 0.788i)4-s + 0.447·5-s + (−1.57 + 0.767i)6-s + 0.433i·7-s + (−0.207 + 0.978i)8-s + 2.06·9-s + (−0.401 + 0.196i)10-s + 0.308·11-s + (1.07 − 1.37i)12-s + 1.52i·13-s + (−0.189 − 0.389i)14-s + 0.783·15-s + (−0.241 − 0.970i)16-s + (−0.815 − 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.735 - 0.677i$
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: \((2,\ 680,\ (\ :1/2),\ 0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83798 + 0.717826i\)
\(L(\frac12)\) \(\approx\) \(1.83798 + 0.717826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.619i)T \)
5 \( 1 - T \)
17 \( 1 + (3.36 + 2.38i)T \)
good3 \( 1 - 3.03T + 3T^{2} \)
7 \( 1 - 1.14iT - 7T^{2} \)
11 \( 1 - 1.02T + 11T^{2} \)
13 \( 1 - 5.48iT - 13T^{2} \)
19 \( 1 - 0.131iT - 19T^{2} \)
23 \( 1 - 2.38iT - 23T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + 0.752iT - 31T^{2} \)
37 \( 1 + 7.71T + 37T^{2} \)
41 \( 1 + 6.10iT - 41T^{2} \)
43 \( 1 + 9.64iT - 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 - 3.72T + 61T^{2} \)
67 \( 1 - 3.72iT - 67T^{2} \)
71 \( 1 + 6.62iT - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 3.85iT - 83T^{2} \)
89 \( 1 + 3.64T + 89T^{2} \)
97 \( 1 - 3.73iT - 97T^{2} \)
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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

−10.04103205688460815064161394401, −9.427009775866678408201081896110, −8.825398587165593262495082515691, −8.371038806224990530260320036460, −7.09320794116043756815155004062, −6.73003515462263592356029259703, −5.20626781339758102599700696096, −3.85105184836070205919290325849, −2.42517821432172883517508192587, −1.79000065380338496340647060319, 1.35387498283910546857077811325, 2.57728818350625818371586100271, 3.29797919670435709469348242343, 4.40113160048506036898879537163, 6.29934739042821732687801487693, 7.33177710770249072747557440735, 8.062404884196703727409362272327, 8.705157062867490512487262479864, 9.380227254272523622462825347591, 10.31266383229105498196927281104

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