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Properties

Label 2-680-17.8-c1-0-12
Degree $2$
Conductor $680$
Sign $0.587 + 0.809i$
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  + (0.424 − 1.02i)3-s + (0.923 + 0.382i)5-s + (−0.208 + 0.0864i)7-s + (1.25 + 1.25i)9-s + (−1.54 − 3.72i)11-s − 4.93i·13-s + (0.783 − 0.783i)15-s + (3.34 + 2.41i)17-s + (2.08 − 2.08i)19-s + 0.250i·21-s + (0.174 + 0.422i)23-s + (0.707 + 0.707i)25-s + (4.88 − 2.02i)27-s + (3.53 + 1.46i)29-s + (−1.42 + 3.43i)31-s + ⋯
L(s)  = 1  + (0.244 − 0.591i)3-s + (0.413 + 0.171i)5-s + (−0.0788 + 0.0326i)7-s + (0.417 + 0.417i)9-s + (−0.465 − 1.12i)11-s − 1.36i·13-s + (0.202 − 0.202i)15-s + (0.810 + 0.585i)17-s + (0.479 − 0.479i)19-s + 0.0546i·21-s + (0.0364 + 0.0880i)23-s + (0.141 + 0.141i)25-s + (0.940 − 0.389i)27-s + (0.655 + 0.271i)29-s + (−0.255 + 0.616i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.587 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58875 - 0.809742i\)
\(L(\frac12)\) \(\approx\) \(1.58875 - 0.809742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-3.34 - 2.41i)T \)
good3 \( 1 + (-0.424 + 1.02i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.208 - 0.0864i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.54 + 3.72i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.93iT - 13T^{2} \)
19 \( 1 + (-2.08 + 2.08i)T - 19iT^{2} \)
23 \( 1 + (-0.174 - 0.422i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.53 - 1.46i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.42 - 3.43i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-2.35 + 5.69i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.17 + 1.31i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.84 + 1.84i)T + 43iT^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 + (1.52 - 1.52i)T - 53iT^{2} \)
59 \( 1 + (2.35 + 2.35i)T + 59iT^{2} \)
61 \( 1 + (9.09 - 3.76i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 3.06T + 67T^{2} \)
71 \( 1 + (-2.04 + 4.94i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-10.2 - 4.23i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.84 - 4.44i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (10.5 - 10.5i)T - 83iT^{2} \)
89 \( 1 - 17.0iT - 89T^{2} \)
97 \( 1 + (-0.668 - 0.276i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.52705473845568384662471005534, −9.546347858237688397187866144491, −8.364168074909719296680345857309, −7.86616777750972142597967019676, −6.90456015369014932445700019968, −5.81890575357830045229806438033, −5.13478371754289134661974787271, −3.47236612179422861147038920380, −2.56870793800824102206000891234, −1.05439705233639089293063307214, 1.58248321257698523435139911420, 2.98822212478365994245211189156, 4.27560443277447468921029576057, 4.89632631279626038579815836925, 6.19879828550368079353451108837, 7.09777412314366055102836855718, 8.020584275607802532968279251408, 9.318587143382166717461612809262, 9.630106053781629240734369935632, 10.29215306271210545274311337672

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