Properties

Label 2-1617-77.76-c1-0-67
Degree $2$
Conductor $1617$
Sign $-0.365 + 0.930i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  + 2.69i·2-s i·3-s − 5.27·4-s − 3.60i·5-s + 2.69·6-s − 8.81i·8-s − 9-s + 9.72·10-s + (−2.32 − 2.36i)11-s + 5.27i·12-s + 4.29·13-s − 3.60·15-s + 13.2·16-s − 6.77·17-s − 2.69i·18-s + 2.03·19-s + ⋯
L(s)  = 1  + 1.90i·2-s − 0.577i·3-s − 2.63·4-s − 1.61i·5-s + 1.10·6-s − 3.11i·8-s − 0.333·9-s + 3.07·10-s + (−0.699 − 0.714i)11-s + 1.52i·12-s + 1.19·13-s − 0.931·15-s + 3.30·16-s − 1.64·17-s − 0.635i·18-s + 0.465·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.365 + 0.930i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2716464002\)
\(L(\frac12)\) \(\approx\) \(0.2716464002\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (2.32 + 2.36i)T \)
good2 \( 1 - 2.69iT - 2T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
13 \( 1 - 4.29T + 13T^{2} \)
17 \( 1 + 6.77T + 17T^{2} \)
19 \( 1 - 2.03T + 19T^{2} \)
23 \( 1 - 1.20T + 23T^{2} \)
29 \( 1 - 0.635iT - 29T^{2} \)
31 \( 1 - 5.92iT - 31T^{2} \)
37 \( 1 + 9.93T + 37T^{2} \)
41 \( 1 + 1.61T + 41T^{2} \)
43 \( 1 - 6.25iT - 43T^{2} \)
47 \( 1 - 0.262iT - 47T^{2} \)
53 \( 1 + 7.12T + 53T^{2} \)
59 \( 1 + 3.79iT - 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 - 6.29T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 5.55T + 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 - 1.67T + 83T^{2} \)
89 \( 1 - 8.75iT - 89T^{2} \)
97 \( 1 + 2.61iT - 97T^{2} \)
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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

−8.832547368996987761369234174151, −8.329644328434269345412442657245, −7.64177781315286869336949575545, −6.63326107436819484242606873716, −6.03130856387672913268202831161, −5.15692367611745614669366126531, −4.70606946840557594080286468231, −3.53451962317713578178456487368, −1.32257559596447586422923106910, −0.11173572224133136144972443325, 1.91787009822172972053565836835, 2.68384560143045458937566934843, 3.48412546148982778942792499439, 4.17607946596231524732187048019, 5.15163220123799145054934721444, 6.28215443613085814605702538449, 7.37130041857328238216212908483, 8.499259738438932632949184627058, 9.189995881543039382144945074827, 10.08254940026568245601790616611

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