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Properties

Label 2-680-680.509-c1-0-101
Degree $2$
Conductor $680$
Sign $0.173 - 0.984i$
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  + (0.483 − 1.32i)2-s − 0.980i·3-s + (−1.53 − 1.28i)4-s + (−2.20 − 0.380i)5-s + (−1.30 − 0.474i)6-s − 2.74·7-s + (−2.44 + 1.41i)8-s + 2.03·9-s + (−1.57 + 2.74i)10-s + 1.05·11-s + (−1.26 + 1.50i)12-s − 2.44·13-s + (−1.32 + 3.64i)14-s + (−0.372 + 2.16i)15-s + (0.694 + 3.93i)16-s + (−2.07 + 3.56i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s − 0.566i·3-s + (−0.766 − 0.642i)4-s + (−0.985 − 0.170i)5-s + (−0.531 − 0.193i)6-s − 1.03·7-s + (−0.866 + 0.499i)8-s + 0.679·9-s + (−0.496 + 0.867i)10-s + 0.318·11-s + (−0.363 + 0.433i)12-s − 0.679·13-s + (−0.354 + 0.975i)14-s + (−0.0962 + 0.557i)15-s + (0.173 + 0.984i)16-s + (−0.503 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.173 - 0.984i$
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: \((2,\ 680,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0681723 + 0.0571972i\)
\(L(\frac12)\) \(\approx\) \(0.0681723 + 0.0571972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.483 + 1.32i)T \)
5 \( 1 + (2.20 + 0.380i)T \)
17 \( 1 + (2.07 - 3.56i)T \)
good3 \( 1 + 0.980iT - 3T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 - 1.05T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
19 \( 1 + 1.36iT - 19T^{2} \)
23 \( 1 + 0.720T + 23T^{2} \)
29 \( 1 + 5.30T + 29T^{2} \)
31 \( 1 - 6.33iT - 31T^{2} \)
37 \( 1 + 2.52iT - 37T^{2} \)
41 \( 1 - 8.39iT - 41T^{2} \)
43 \( 1 + 8.91T + 43T^{2} \)
47 \( 1 + 6.52iT - 47T^{2} \)
53 \( 1 + 0.0463T + 53T^{2} \)
59 \( 1 + 7.30iT - 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 + 8.22T + 67T^{2} \)
71 \( 1 + 7.91iT - 71T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 + 8.93iT - 79T^{2} \)
83 \( 1 + 7.61T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.941582480354551744102865400090, −9.130628090403649166949878948197, −8.171238460894600852176182079358, −7.07054312194855577095816181845, −6.31520452254754314504373671947, −4.90930268575942591407058811350, −3.98858016818569030678345029854, −3.11181309017895146935503465732, −1.65404126817085001824070241055, −0.04223658646663795998103764071, 3.03759888588052639801022908072, 3.97478340133385580129812206640, 4.62723444407761094260488797968, 5.82877203060304774546338120901, 7.01718445812403571621758972096, 7.30400726077802236176423831445, 8.492895313200754889448958137105, 9.463632900308659755274241508356, 9.946932575888683356346926541173, 11.22504345116257325066932650084

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