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Properties

Label 2-680-136.101-c1-0-65
Degree $2$
Conductor $680$
Sign $-0.922 + 0.386i$
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  + (1.02 − 0.971i)2-s − 0.744·3-s + (0.113 − 1.99i)4-s + 5-s + (−0.765 + 0.723i)6-s + 0.964i·7-s + (−1.82 − 2.16i)8-s − 2.44·9-s + (1.02 − 0.971i)10-s − 4.79·11-s + (−0.0847 + 1.48i)12-s − 5.30i·13-s + (0.936 + 0.991i)14-s − 0.744·15-s + (−3.97 − 0.454i)16-s + (1.23 − 3.93i)17-s + ⋯
L(s)  = 1  + (0.726 − 0.686i)2-s − 0.429·3-s + (0.0568 − 0.998i)4-s + 0.447·5-s + (−0.312 + 0.295i)6-s + 0.364i·7-s + (−0.644 − 0.764i)8-s − 0.815·9-s + (0.325 − 0.307i)10-s − 1.44·11-s + (−0.0244 + 0.429i)12-s − 1.47i·13-s + (0.250 + 0.265i)14-s − 0.192·15-s + (−0.993 − 0.113i)16-s + (0.298 − 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.922 + 0.386i$
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: \((2,\ 680,\ (\ :1/2),\ -0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264936 - 1.31846i\)
\(L(\frac12)\) \(\approx\) \(0.264936 - 1.31846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 + 0.971i)T \)
5 \( 1 - T \)
17 \( 1 + (-1.23 + 3.93i)T \)
good3 \( 1 + 0.744T + 3T^{2} \)
7 \( 1 - 0.964iT - 7T^{2} \)
11 \( 1 + 4.79T + 11T^{2} \)
13 \( 1 + 5.30iT - 13T^{2} \)
19 \( 1 + 6.23iT - 19T^{2} \)
23 \( 1 - 3.62iT - 23T^{2} \)
29 \( 1 - 7.67T + 29T^{2} \)
31 \( 1 + 3.78iT - 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 0.571iT - 41T^{2} \)
43 \( 1 - 8.74iT - 43T^{2} \)
47 \( 1 - 6.19T + 47T^{2} \)
53 \( 1 + 4.27iT - 53T^{2} \)
59 \( 1 - 11.5iT - 59T^{2} \)
61 \( 1 + 4.51T + 61T^{2} \)
67 \( 1 + 3.12iT - 67T^{2} \)
71 \( 1 + 7.08iT - 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 - 7.59iT - 79T^{2} \)
83 \( 1 - 1.74iT - 83T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 - 5.26iT - 97T^{2} \)
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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

−10.46804139565046874460704115870, −9.518797314363830190134077595472, −8.526827572871021289203440896192, −7.34925330476234405144346448190, −6.09437462887772522940871110936, −5.32193313458346249568553086283, −4.94890797289343508543157762809, −3.02042137912313787196666874510, −2.60911203417290371003694125628, −0.55636054294359449171420092391, 2.20909605131064171720624361654, 3.51204517384432342811891485022, 4.68048626017197770499511705845, 5.53922848830841497061698920456, 6.26401485959784233647500522719, 7.12217921793714992233422815775, 8.237680061091272793325441404514, 8.781222206957959370665939114443, 10.29383556191697828617667135606, 10.76340639083899047417839928546

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