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Properties

Label 2-680-680.509-c1-0-84
Degree $2$
Conductor $680$
Sign $-0.922 + 0.384i$
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  + (−1.36 + 0.377i)2-s − 0.944i·3-s + (1.71 − 1.02i)4-s + (0.0976 − 2.23i)5-s + (0.356 + 1.28i)6-s − 2.88·7-s + (−1.94 + 2.05i)8-s + 2.10·9-s + (0.710 + 3.08i)10-s − 2.86·11-s + (−0.972 − 1.61i)12-s + 3.06·13-s + (3.93 − 1.09i)14-s + (−2.10 − 0.0922i)15-s + (1.87 − 3.53i)16-s + (1.50 − 3.84i)17-s + ⋯
L(s)  = 1  + (−0.963 + 0.267i)2-s − 0.545i·3-s + (0.857 − 0.514i)4-s + (0.0436 − 0.999i)5-s + (0.145 + 0.525i)6-s − 1.09·7-s + (−0.688 + 0.725i)8-s + 0.702·9-s + (0.224 + 0.974i)10-s − 0.865·11-s + (−0.280 − 0.467i)12-s + 0.850·13-s + (1.05 − 0.291i)14-s + (−0.544 − 0.0238i)15-s + (0.469 − 0.882i)16-s + (0.363 − 0.931i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.922 + 0.384i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.922 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102517 - 0.512117i\)
\(L(\frac12)\) \(\approx\) \(0.102517 - 0.512117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.377i)T \)
5 \( 1 + (-0.0976 + 2.23i)T \)
17 \( 1 + (-1.50 + 3.84i)T \)
good3 \( 1 + 0.944iT - 3T^{2} \)
7 \( 1 + 2.88T + 7T^{2} \)
11 \( 1 + 2.86T + 11T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 - 3.49iT - 31T^{2} \)
37 \( 1 - 7.07iT - 37T^{2} \)
41 \( 1 + 4.23iT - 41T^{2} \)
43 \( 1 + 4.28T + 43T^{2} \)
47 \( 1 - 0.586iT - 47T^{2} \)
53 \( 1 + 13.9T + 53T^{2} \)
59 \( 1 - 7.50iT - 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 + 7.98iT - 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 5.19T + 97T^{2} \)
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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

−9.825035410312794449819294219921, −9.323489326312091693292876456197, −8.333492135787434121383016298951, −7.61009095765210246867826606934, −6.70666214938179093128194713251, −5.90339712121450814315870513141, −4.79209922052341194325946775813, −3.12819480445523404869977710863, −1.69148499810511319790718379359, −0.36667894487135018956446237612, 1.92647384158513698695825470545, 3.31038012299531204959335985527, 3.85604988771362063864679147927, 5.89186771553388683863997139795, 6.48164371205840927749166364695, 7.56399477619580704608072894337, 8.230566229654759249741914883496, 9.634812669896409955812374964389, 9.885972352465344976209684793146, 10.65751950010327109220787116922

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