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Properties

Label 2-680-136.101-c1-0-10
Degree $2$
Conductor $680$
Sign $0.356 - 0.934i$
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  + (−1.40 − 0.124i)2-s − 0.904·3-s + (1.96 + 0.351i)4-s − 5-s + (1.27 + 0.112i)6-s − 1.75i·7-s + (−2.72 − 0.739i)8-s − 2.18·9-s + (1.40 + 0.124i)10-s + 0.717·11-s + (−1.78 − 0.317i)12-s + 2.57i·13-s + (−0.219 + 2.47i)14-s + 0.904·15-s + (3.75 + 1.38i)16-s + (0.409 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0881i)2-s − 0.522·3-s + (0.984 + 0.175i)4-s − 0.447·5-s + (0.520 + 0.0460i)6-s − 0.664i·7-s + (−0.965 − 0.261i)8-s − 0.727·9-s + (0.445 + 0.0394i)10-s + 0.216·11-s + (−0.514 − 0.0916i)12-s + 0.713i·13-s + (−0.0585 + 0.661i)14-s + 0.233·15-s + (0.938 + 0.345i)16-s + (0.0992 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.356 - 0.934i$
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: \((2,\ 680,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394910 + 0.272123i\)
\(L(\frac12)\) \(\approx\) \(0.394910 + 0.272123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.124i)T \)
5 \( 1 + T \)
17 \( 1 + (-0.409 + 4.10i)T \)
good3 \( 1 + 0.904T + 3T^{2} \)
7 \( 1 + 1.75iT - 7T^{2} \)
11 \( 1 - 0.717T + 11T^{2} \)
13 \( 1 - 2.57iT - 13T^{2} \)
19 \( 1 - 6.22iT - 19T^{2} \)
23 \( 1 + 2.65iT - 23T^{2} \)
29 \( 1 - 2.07T + 29T^{2} \)
31 \( 1 - 5.63iT - 31T^{2} \)
37 \( 1 + 7.18T + 37T^{2} \)
41 \( 1 - 5.90iT - 41T^{2} \)
43 \( 1 - 8.86iT - 43T^{2} \)
47 \( 1 - 4.38T + 47T^{2} \)
53 \( 1 + 2.07iT - 53T^{2} \)
59 \( 1 - 4.37iT - 59T^{2} \)
61 \( 1 - 9.54T + 61T^{2} \)
67 \( 1 - 14.7iT - 67T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 - 6.23iT - 73T^{2} \)
79 \( 1 + 3.03iT - 79T^{2} \)
83 \( 1 - 9.62iT - 83T^{2} \)
89 \( 1 - 1.67T + 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−10.56468720553207677448276649399, −9.933706662931869483303643880597, −8.848394486233746516886991092054, −8.178979860126299697402331220544, −7.15870718844428287393345287792, −6.51652593698073617856007379791, −5.39776026738357876637953293458, −4.02556689611233604479717795232, −2.80987093311755172215068959367, −1.13236398133307562214949741248, 0.43381678444069073290755532872, 2.26482488539710112492016002007, 3.45699863344374346265984899493, 5.19674341964244593677104102032, 5.93877360726127606485106450838, 6.86033273869514026111233965424, 7.86645073548582332857778910491, 8.677726330576760551734493594768, 9.250126246945219191865090923940, 10.52714465863865779114676465720

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