Properties

Label 2-1617-231.230-c0-0-2
Degree $2$
Conductor $1617$
Sign $0.233 - 0.972i$
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  + (−0.382 + 0.923i)3-s − 4-s + 1.84·5-s + (−0.707 − 0.707i)9-s + i·11-s + (0.382 − 0.923i)12-s + (−0.707 + 1.70i)15-s + 16-s − 1.84·20-s + 1.41i·23-s + 2.41·25-s + (0.923 − 0.382i)27-s − 0.765i·31-s + (−0.923 − 0.382i)33-s + (0.707 + 0.707i)36-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s − 4-s + 1.84·5-s + (−0.707 − 0.707i)9-s + i·11-s + (0.382 − 0.923i)12-s + (−0.707 + 1.70i)15-s + 16-s − 1.84·20-s + 1.41i·23-s + 2.41·25-s + (0.923 − 0.382i)27-s − 0.765i·31-s + (−0.923 − 0.382i)33-s + (0.707 + 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1616, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.233 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046549088\)
\(L(\frac12)\) \(\approx\) \(1.046549088\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + T^{2} \)
5 \( 1 - 1.84T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 0.765iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 0.765T + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - 0.765T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.765T + T^{2} \)
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

−9.771170037108966007770384612620, −9.303812252300225797753710115639, −8.582351810475581580337852752789, −7.29967814098500318214972648886, −6.13553805900387566812368783370, −5.57034218105287184214200374641, −4.92113006035103134806694209061, −4.10683777032202789034677942801, −2.88828605168723198900361559424, −1.54785031717955117539781256490, 0.983409077516889507731898657844, 2.13347694220110486998580563407, 3.20184263750341287248936640681, 4.81952138589266903924196226809, 5.41805485894364728721709838285, 6.18069387153598171444242559877, 6.68562381780977312985875891172, 8.007415759038859089344953418159, 8.720230768368919122572537044813, 9.260578343396553424189560883498

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