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Properties

Label 2-680-85.4-c1-0-9
Degree $2$
Conductor $680$
Sign $0.719 - 0.695i$
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  + (0.731 + 0.731i)3-s + (−2.20 + 0.357i)5-s + (1.16 − 1.16i)7-s − 1.92i·9-s + (2.68 + 2.68i)11-s + 3.47i·13-s + (−1.87 − 1.35i)15-s + (4.11 + 0.227i)17-s + 6.84i·19-s + 1.70·21-s + (5.10 − 5.10i)23-s + (4.74 − 1.58i)25-s + (3.60 − 3.60i)27-s + (−3.39 + 3.39i)29-s + (−1.87 + 1.87i)31-s + ⋯
L(s)  = 1  + (0.422 + 0.422i)3-s + (−0.987 + 0.160i)5-s + (0.441 − 0.441i)7-s − 0.642i·9-s + (0.809 + 0.809i)11-s + 0.964i·13-s + (−0.484 − 0.349i)15-s + (0.998 + 0.0552i)17-s + 1.56i·19-s + 0.373·21-s + (1.06 − 1.06i)23-s + (0.948 − 0.316i)25-s + (0.694 − 0.694i)27-s + (−0.629 + 0.629i)29-s + (−0.336 + 0.336i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.719 - 0.695i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.719 - 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49327 + 0.603746i\)
\(L(\frac12)\) \(\approx\) \(1.49327 + 0.603746i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 0.357i)T \)
17 \( 1 + (-4.11 - 0.227i)T \)
good3 \( 1 + (-0.731 - 0.731i)T + 3iT^{2} \)
7 \( 1 + (-1.16 + 1.16i)T - 7iT^{2} \)
11 \( 1 + (-2.68 - 2.68i)T + 11iT^{2} \)
13 \( 1 - 3.47iT - 13T^{2} \)
19 \( 1 - 6.84iT - 19T^{2} \)
23 \( 1 + (-5.10 + 5.10i)T - 23iT^{2} \)
29 \( 1 + (3.39 - 3.39i)T - 29iT^{2} \)
31 \( 1 + (1.87 - 1.87i)T - 31iT^{2} \)
37 \( 1 + (-2.43 - 2.43i)T + 37iT^{2} \)
41 \( 1 + (-8.54 - 8.54i)T + 41iT^{2} \)
43 \( 1 + 7.78T + 43T^{2} \)
47 \( 1 + 5.02iT - 47T^{2} \)
53 \( 1 - 4.48T + 53T^{2} \)
59 \( 1 + 12.9iT - 59T^{2} \)
61 \( 1 + (3.33 + 3.33i)T + 61iT^{2} \)
67 \( 1 - 2.91iT - 67T^{2} \)
71 \( 1 + (6.72 - 6.72i)T - 71iT^{2} \)
73 \( 1 + (7.27 + 7.27i)T + 73iT^{2} \)
79 \( 1 + (5.70 + 5.70i)T + 79iT^{2} \)
83 \( 1 - 7.50T + 83T^{2} \)
89 \( 1 - 5.35T + 89T^{2} \)
97 \( 1 + (-9.65 - 9.65i)T + 97iT^{2} \)
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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

−10.52972642400861807510625572116, −9.691518277796898274419472851932, −8.892843098058808247121110311271, −8.010799887590794536310366326674, −7.16041097844326887248672829319, −6.34829973584821471290312163769, −4.73574088973279206694641200984, −4.02759811943717523194062516009, −3.25138769725563808590519879981, −1.39207308711954001883218274629, 0.996882918495336536058647858678, 2.68062576972839379872355487933, 3.64225963930961107692149692806, 4.93835677836780590762189218422, 5.76972443956122733625994640669, 7.30347691150212205576736865750, 7.64964328733664123168682956541, 8.652455390618065728541959837948, 9.153994178615563787709375459612, 10.61190714966204876092510522427

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