Properties

Label 2-1617-77.76-c1-0-28
Degree $2$
Conductor $1617$
Sign $0.0422 - 0.999i$
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  + 2.50i·2-s i·3-s − 4.26·4-s − 0.980i·5-s + 2.50·6-s − 5.66i·8-s − 9-s + 2.45·10-s + (−2.59 + 2.06i)11-s + 4.26i·12-s − 0.941·13-s − 0.980·15-s + 5.66·16-s + 5.17·17-s − 2.50i·18-s + 2.54·19-s + ⋯
L(s)  = 1  + 1.76i·2-s − 0.577i·3-s − 2.13·4-s − 0.438i·5-s + 1.02·6-s − 2.00i·8-s − 0.333·9-s + 0.776·10-s + (−0.782 + 0.622i)11-s + 1.23i·12-s − 0.261·13-s − 0.253·15-s + 1.41·16-s + 1.25·17-s − 0.589i·18-s + 0.584·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.0422 - 0.999i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.0422 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.345839603\)
\(L(\frac12)\) \(\approx\) \(1.345839603\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.06i)T \)
good2 \( 1 - 2.50iT - 2T^{2} \)
5 \( 1 + 0.980iT - 5T^{2} \)
13 \( 1 + 0.941T + 13T^{2} \)
17 \( 1 - 5.17T + 17T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 + 0.758T + 23T^{2} \)
29 \( 1 + 8.74iT - 29T^{2} \)
31 \( 1 + 6.36iT - 31T^{2} \)
37 \( 1 - 2.98T + 37T^{2} \)
41 \( 1 - 3.80T + 41T^{2} \)
43 \( 1 - 6.73iT - 43T^{2} \)
47 \( 1 - 9.73iT - 47T^{2} \)
53 \( 1 + 4.49T + 53T^{2} \)
59 \( 1 - 14.7iT - 59T^{2} \)
61 \( 1 - 5.62T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 - 7.65iT - 79T^{2} \)
83 \( 1 - 6.28T + 83T^{2} \)
89 \( 1 - 2.38iT - 89T^{2} \)
97 \( 1 + 7.30iT - 97T^{2} \)
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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

−9.534743729005041611119777634970, −8.277815783533588919089612127572, −7.83313467984304691711702635376, −7.37254638394051629837523296757, −6.37182995827120214838618821756, −5.68357518736762649004507492001, −4.99997235024326447022898221692, −4.13410407157619879956409909393, −2.59572142674397329954461377852, −0.809641683190477001243046158215, 0.827932144404873918686973540428, 2.24270454378622121159147153084, 3.31890088150033406342332847832, 3.50166629872219475487612268555, 5.01231332028317944959856678591, 5.31719861972371001824695525901, 6.81123015048835459357978069750, 7.989673458043024979696006052803, 8.740371369315619630224699671633, 9.575114397020370541854072104163

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