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Properties

Label 2-680-136.101-c1-0-26
Degree $2$
Conductor $680$
Sign $0.648 + 0.761i$
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  + (−0.0976 − 1.41i)2-s − 0.700·3-s + (−1.98 + 0.275i)4-s + 5-s + (0.0683 + 0.987i)6-s + 0.472i·7-s + (0.582 + 2.76i)8-s − 2.50·9-s + (−0.0976 − 1.41i)10-s + 3.76·11-s + (1.38 − 0.192i)12-s + 2.08i·13-s + (0.666 − 0.0461i)14-s − 0.700·15-s + (3.84 − 1.09i)16-s + (3.62 − 1.96i)17-s + ⋯
L(s)  = 1  + (−0.0690 − 0.997i)2-s − 0.404·3-s + (−0.990 + 0.137i)4-s + 0.447·5-s + (0.0279 + 0.403i)6-s + 0.178i·7-s + (0.205 + 0.978i)8-s − 0.836·9-s + (−0.0308 − 0.446i)10-s + 1.13·11-s + (0.400 − 0.0556i)12-s + 0.578i·13-s + (0.178 − 0.0123i)14-s − 0.180·15-s + (0.962 − 0.272i)16-s + (0.878 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.648 + 0.761i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.648 + 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11941 - 0.517250i\)
\(L(\frac12)\) \(\approx\) \(1.11941 - 0.517250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0976 + 1.41i)T \)
5 \( 1 - T \)
17 \( 1 + (-3.62 + 1.96i)T \)
good3 \( 1 + 0.700T + 3T^{2} \)
7 \( 1 - 0.472iT - 7T^{2} \)
11 \( 1 - 3.76T + 11T^{2} \)
13 \( 1 - 2.08iT - 13T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 + 3.04iT - 23T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 - 1.89iT - 31T^{2} \)
37 \( 1 - 4.90T + 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 8.18iT - 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 + 6.11iT - 53T^{2} \)
59 \( 1 - 5.25iT - 59T^{2} \)
61 \( 1 - 4.08T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + 11.7iT - 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 + 1.62iT - 83T^{2} \)
89 \( 1 - 4.13T + 89T^{2} \)
97 \( 1 - 1.51iT - 97T^{2} \)
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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

−10.41346904576042203446779244688, −9.637652148187153475963211256711, −8.878638095763810295505797006408, −8.100207756279724459182056412786, −6.61791298286122530737707925600, −5.74599930823346573951520653174, −4.77428225815414443889302573969, −3.63316532591209486568052488692, −2.48824287346367331534487878398, −1.10395103116556398355373740413, 0.970454527534283414718490971525, 3.12845114393485068055028037996, 4.42249102587987894724960568012, 5.45169800984947661229780884299, 6.14536121560195538556739163605, 6.89433699724591045292795547561, 8.002095987186155226515023058948, 8.773280833815858354694090273878, 9.611627672253143434560938781917, 10.39979554760637143358597306859

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