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Properties

Label 2-680-17.9-c1-0-8
Degree $2$
Conductor $680$
Sign $0.654 + 0.756i$
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  + (−0.465 − 0.192i)3-s + (−0.382 + 0.923i)5-s + (0.134 + 0.324i)7-s + (−1.94 − 1.94i)9-s + (1.41 − 0.587i)11-s − 3.28i·13-s + (0.355 − 0.355i)15-s + (3.96 − 1.11i)17-s + (0.386 − 0.386i)19-s − 0.176i·21-s + (3.21 − 1.33i)23-s + (−0.707 − 0.707i)25-s + (1.10 + 2.67i)27-s + (2.60 − 6.28i)29-s + (6.87 + 2.84i)31-s + ⋯
L(s)  = 1  + (−0.268 − 0.111i)3-s + (−0.171 + 0.413i)5-s + (0.0507 + 0.122i)7-s + (−0.647 − 0.647i)9-s + (0.427 − 0.177i)11-s − 0.911i·13-s + (0.0919 − 0.0919i)15-s + (0.962 − 0.270i)17-s + (0.0886 − 0.0886i)19-s − 0.0385i·21-s + (0.670 − 0.277i)23-s + (−0.141 − 0.141i)25-s + (0.213 + 0.514i)27-s + (0.483 − 1.16i)29-s + (1.23 + 0.511i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.654 + 0.756i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.654 + 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13359 - 0.518192i\)
\(L(\frac12)\) \(\approx\) \(1.13359 - 0.518192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 + (-3.96 + 1.11i)T \)
good3 \( 1 + (0.465 + 0.192i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.134 - 0.324i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.41 + 0.587i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.28iT - 13T^{2} \)
19 \( 1 + (-0.386 + 0.386i)T - 19iT^{2} \)
23 \( 1 + (-3.21 + 1.33i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.60 + 6.28i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-6.87 - 2.84i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.970 - 0.402i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.85 + 9.31i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.396 - 0.396i)T + 43iT^{2} \)
47 \( 1 - 5.86iT - 47T^{2} \)
53 \( 1 + (0.763 - 0.763i)T - 53iT^{2} \)
59 \( 1 + (9.56 + 9.56i)T + 59iT^{2} \)
61 \( 1 + (0.893 + 2.15i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 8.71T + 67T^{2} \)
71 \( 1 + (2.21 + 0.915i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.23 + 7.81i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.823 - 0.341i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (9.03 - 9.03i)T - 83iT^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 + (0.917 - 2.21i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.42117450245964813665710415579, −9.571315939143231234140072847021, −8.584111930442831989263643956204, −7.79010988187211267059554685651, −6.71384976030215447640595802885, −5.96424981440146836896579883148, −5.01830243477487297211804336017, −3.59953045953642660650574595089, −2.75125605512149169614727535573, −0.78241234524996648549508751393, 1.36560116283714401256189873367, 2.94526620418084235793676221966, 4.26245248501292047591617913125, 5.09303953352444606395716301015, 6.07120087696183919045836759540, 7.10162983458235372960999891565, 8.088226479794055836543076018817, 8.840027415734085459221458049538, 9.768611475879854607343195961114, 10.63111205415699662685170792196

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