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Properties

Label 2-680-17.9-c1-0-7
Degree $2$
Conductor $680$
Sign $0.453 - 0.891i$
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  + (2.36 + 0.978i)3-s + (−0.382 + 0.923i)5-s + (0.904 + 2.18i)7-s + (2.49 + 2.49i)9-s + (2.93 − 1.21i)11-s + 1.34i·13-s + (−1.80 + 1.80i)15-s + (−2.53 − 3.25i)17-s + (−0.401 + 0.401i)19-s + 6.04i·21-s + (−0.746 + 0.309i)23-s + (−0.707 − 0.707i)25-s + (0.522 + 1.26i)27-s + (−2.37 + 5.74i)29-s + (2.29 + 0.951i)31-s + ⋯
L(s)  = 1  + (1.36 + 0.564i)3-s + (−0.171 + 0.413i)5-s + (0.341 + 0.825i)7-s + (0.833 + 0.833i)9-s + (0.883 − 0.366i)11-s + 0.373i·13-s + (−0.466 + 0.466i)15-s + (−0.613 − 0.789i)17-s + (−0.0920 + 0.0920i)19-s + 1.31i·21-s + (−0.155 + 0.0644i)23-s + (−0.141 − 0.141i)25-s + (0.100 + 0.242i)27-s + (−0.441 + 1.06i)29-s + (0.412 + 0.170i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.453 - 0.891i$
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: \((2,\ 680,\ (\ :1/2),\ 0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03323 + 1.24660i\)
\(L(\frac12)\) \(\approx\) \(2.03323 + 1.24660i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 + 3.25i)T \)
good3 \( 1 + (-2.36 - 0.978i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.904 - 2.18i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2.93 + 1.21i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.34iT - 13T^{2} \)
19 \( 1 + (0.401 - 0.401i)T - 19iT^{2} \)
23 \( 1 + (0.746 - 0.309i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (2.37 - 5.74i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.29 - 0.951i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.519 + 0.215i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.35 - 3.27i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.64 - 1.64i)T + 43iT^{2} \)
47 \( 1 + 4.11iT - 47T^{2} \)
53 \( 1 + (-8.56 + 8.56i)T - 53iT^{2} \)
59 \( 1 + (7.93 + 7.93i)T + 59iT^{2} \)
61 \( 1 + (-2.70 - 6.52i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + (3.44 + 1.42i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-6.20 + 14.9i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.51 + 0.627i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-5.55 + 5.55i)T - 83iT^{2} \)
89 \( 1 - 5.11iT - 89T^{2} \)
97 \( 1 + (4.79 - 11.5i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.51839546705954922487129599164, −9.442683766489577108029222177004, −8.973327874005628917121008534937, −8.320354866274078054074790605343, −7.29223572820788439297240094854, −6.28328462658129869942001236510, −4.95492706905081741701322771935, −3.89883428803706446903963467149, −3.01499564225432757736748957029, −1.99290798813182894782382871320, 1.25601327323986829941048734533, 2.40324070326566013055775794882, 3.79382881731437716119012086385, 4.40576224612476217381401352598, 6.03053634147389117563246529319, 7.16583138684510218592632568544, 7.75540902876010259377777221214, 8.578369202167777174565640810046, 9.212121586869809807663859632717, 10.19228649410223874017290196538

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