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Properties

Label 2-680-136.101-c1-0-31
Degree $2$
Conductor $680$
Sign $0.814 + 0.580i$
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  + (−0.0399 + 1.41i)2-s − 2.61·3-s + (−1.99 − 0.113i)4-s + 5-s + (0.104 − 3.69i)6-s + 3.54i·7-s + (0.239 − 2.81i)8-s + 3.84·9-s + (−0.0399 + 1.41i)10-s − 6.42·11-s + (5.22 + 0.295i)12-s − 2.23i·13-s + (−5.01 − 0.141i)14-s − 2.61·15-s + (3.97 + 0.451i)16-s + (−2.09 + 3.54i)17-s + ⋯
L(s)  = 1  + (−0.0282 + 0.999i)2-s − 1.51·3-s + (−0.998 − 0.0565i)4-s + 0.447·5-s + (0.0427 − 1.51i)6-s + 1.34i·7-s + (0.0847 − 0.996i)8-s + 1.28·9-s + (−0.0126 + 0.447i)10-s − 1.93·11-s + (1.50 + 0.0853i)12-s − 0.619i·13-s + (−1.34 − 0.0379i)14-s − 0.675·15-s + (0.993 + 0.112i)16-s + (−0.509 + 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.814 + 0.580i$
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: \((2,\ 680,\ (\ :1/2),\ 0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252333 - 0.0806949i\)
\(L(\frac12)\) \(\approx\) \(0.252333 - 0.0806949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0399 - 1.41i)T \)
5 \( 1 - T \)
17 \( 1 + (2.09 - 3.54i)T \)
good3 \( 1 + 2.61T + 3T^{2} \)
7 \( 1 - 3.54iT - 7T^{2} \)
11 \( 1 + 6.42T + 11T^{2} \)
13 \( 1 + 2.23iT - 13T^{2} \)
19 \( 1 + 0.717iT - 19T^{2} \)
23 \( 1 + 3.87iT - 23T^{2} \)
29 \( 1 + 1.66T + 29T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 - 9.42T + 37T^{2} \)
41 \( 1 + 9.35iT - 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 - 7.93T + 47T^{2} \)
53 \( 1 + 6.51iT - 53T^{2} \)
59 \( 1 + 6.56iT - 59T^{2} \)
61 \( 1 + 6.75T + 61T^{2} \)
67 \( 1 + 6.25iT - 67T^{2} \)
71 \( 1 - 9.73iT - 71T^{2} \)
73 \( 1 + 0.392iT - 73T^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 - 4.75iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−10.45697923232640829394134390669, −9.567597936765217556661598294067, −8.487456198590062934046811009081, −7.72976016038733883896251108578, −6.50331917120683382760486634343, −5.67996825310082954225034574612, −5.50465575102669113384470646447, −4.50795216037942074491636635009, −2.51231041544538864314722580199, −0.20045466416331419656324043599, 1.12048013348833204967666648353, 2.73430596364199569668183188068, 4.28179202497328442493192043723, 4.99876042521869111106377568398, 5.77948867704831753741475824831, 7.04257935160507655988842219359, 7.85739830208419922158392668953, 9.298885104617462915951437787615, 10.18882375642128364665762526711, 10.66661559928224917713662685800

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