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Properties

Label 2-1617-231.131-c0-0-10
Degree $2$
Conductor $1617$
Sign $0.532 + 0.846i$
Analytic cond. $0.806988$
Root an. cond. $0.898325$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.608 − 0.793i)3-s + (0.5 + 0.866i)4-s + (0.923 − 1.60i)5-s + (−0.258 − 0.965i)9-s + (−0.866 + 0.5i)11-s + (0.991 + 0.130i)12-s + (−0.707 − 1.70i)15-s + (−0.499 + 0.866i)16-s + 1.84·20-s + (1.22 + 0.707i)23-s + (−1.20 − 2.09i)25-s + (−0.923 − 0.382i)27-s + (−0.662 + 0.382i)31-s + (−0.130 + 0.991i)33-s + (0.707 − 0.707i)36-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)3-s + (0.5 + 0.866i)4-s + (0.923 − 1.60i)5-s + (−0.258 − 0.965i)9-s + (−0.866 + 0.5i)11-s + (0.991 + 0.130i)12-s + (−0.707 − 1.70i)15-s + (−0.499 + 0.866i)16-s + 1.84·20-s + (1.22 + 0.707i)23-s + (−1.20 − 2.09i)25-s + (−0.923 − 0.382i)27-s + (−0.662 + 0.382i)31-s + (−0.130 + 0.991i)33-s + (0.707 − 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.532 + 0.846i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.532 + 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.667995000\)
\(L(\frac12)\) \(\approx\) \(1.667995000\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 \)
11 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−9.131329985881993528074230530280, −8.708401267029468781116273929640, −7.86823996826607998502037325138, −7.30711768031437002214646416624, −6.30001588498620475184922415561, −5.39231950722696926468317575874, −4.50105949069139363015064227452, −3.22684071291148735684081949016, −2.24865603418940823619950729988, −1.38920050538455677501457586007, 2.02685691618754861428873131804, 2.71490729072239713136450942390, 3.44547248685954171994677787752, 4.98513620724119079866608553483, 5.61988813628757992310770358825, 6.49216164257972695537094685676, 7.16560435277797397627340855693, 8.139902807339273977934841188164, 9.273837265905210945225649324280, 9.829374427648097829031666135643

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