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Properties

Label 2-1617-77.76-c1-0-78
Degree $2$
Conductor $1617$
Sign $-0.806 - 0.590i$
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  − 1.03i·2-s − i·3-s + 0.934·4-s − 3.55i·5-s − 1.03·6-s − 3.02i·8-s − 9-s − 3.66·10-s + (0.693 − 3.24i)11-s − 0.934i·12-s − 4.92·13-s − 3.55·15-s − 1.25·16-s − 5.73·17-s + 1.03i·18-s + 2.80·19-s + ⋯
L(s)  = 1  − 0.729i·2-s − 0.577i·3-s + 0.467·4-s − 1.58i·5-s − 0.421·6-s − 1.07i·8-s − 0.333·9-s − 1.16·10-s + (0.208 − 0.977i)11-s − 0.269i·12-s − 1.36·13-s − 0.917·15-s − 0.314·16-s − 1.39·17-s + 0.243i·18-s + 0.644·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.806 - 0.590i$
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: \((2,\ 1617,\ (\ :1/2),\ -0.806 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.605751294\)
\(L(\frac12)\) \(\approx\) \(1.605751294\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.693 + 3.24i)T \)
good2 \( 1 + 1.03iT - 2T^{2} \)
5 \( 1 + 3.55iT - 5T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 - 2.80T + 19T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 - 9.68iT - 29T^{2} \)
31 \( 1 - 7.44iT - 31T^{2} \)
37 \( 1 - 0.0589T + 37T^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 + 6.43iT - 43T^{2} \)
47 \( 1 + 8.08iT - 47T^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + 9.88iT - 59T^{2} \)
61 \( 1 - 6.54T + 61T^{2} \)
67 \( 1 - 5.87T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 4.35T + 73T^{2} \)
79 \( 1 - 6.11iT - 79T^{2} \)
83 \( 1 - 0.336T + 83T^{2} \)
89 \( 1 + 3.49iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−8.975241433592889082919801412072, −8.352307607821875877809963443278, −7.17509961824828067778808279205, −6.72514316533520098250281419682, −5.38470105995850477066263477522, −4.87633450272247174499552706075, −3.59170159685774284169851936702, −2.57164406953748809328463618937, −1.49043260215063097642588420844, −0.59193477990087944142002751729, 2.44842651589244965796297133645, 2.63037575708999293626844171883, 4.14505209654065384085743643039, 4.99119278050511603316968739686, 6.10260490311291502916236881569, 6.71107615666216210669886504285, 7.39928741555457081953561010963, 7.84540777539618937992132387212, 9.309771189397277602277734207806, 9.761601887295446314486299412165

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