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Properties

Label 2-17-17.16-c3-0-2
Degree $2$
Conductor $17$
Sign $-0.174 + 0.984i$
Analytic cond. $1.00303$
Root an. cond. $1.00151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.37·2-s − 7.36i·3-s + 3.37·4-s − 10.1i·5-s + 24.8i·6-s + 17.4i·7-s + 15.6·8-s − 27.2·9-s + 34.0i·10-s − 51.5i·11-s − 24.8i·12-s + 75.2·13-s − 58.9i·14-s − 74.4·15-s − 79.6·16-s + (−12.2 + 69.0i)17-s + ⋯
L(s)  = 1  − 1.19·2-s − 1.41i·3-s + 0.421·4-s − 0.903i·5-s + 1.68i·6-s + 0.943i·7-s + 0.689·8-s − 1.00·9-s + 1.07i·10-s − 1.41i·11-s − 0.597i·12-s + 1.60·13-s − 1.12i·14-s − 1.28·15-s − 1.24·16-s + (−0.174 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.370371 - 0.441653i\)
\(L(\frac12)\) \(\approx\) \(0.370371 - 0.441653i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (12.2 - 69.0i)T \)
good2 \( 1 + 3.37T + 8T^{2} \)
3 \( 1 + 7.36iT - 27T^{2} \)
5 \( 1 + 10.1iT - 125T^{2} \)
7 \( 1 - 17.4iT - 343T^{2} \)
11 \( 1 + 51.5iT - 1.33e3T^{2} \)
13 \( 1 - 75.2T + 2.19e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 + 19.1iT - 1.21e4T^{2} \)
29 \( 1 - 70.7iT - 2.43e4T^{2} \)
31 \( 1 - 41.4iT - 2.97e4T^{2} \)
37 \( 1 - 135. iT - 5.06e4T^{2} \)
41 \( 1 - 288. iT - 6.89e4T^{2} \)
43 \( 1 - 88.2T + 7.95e4T^{2} \)
47 \( 1 - 157.T + 1.03e5T^{2} \)
53 \( 1 - 120.T + 1.48e5T^{2} \)
59 \( 1 + 696.T + 2.05e5T^{2} \)
61 \( 1 + 683. iT - 2.26e5T^{2} \)
67 \( 1 - 123.T + 3.00e5T^{2} \)
71 \( 1 - 225. iT - 3.57e5T^{2} \)
73 \( 1 - 919. iT - 3.89e5T^{2} \)
79 \( 1 + 354. iT - 4.93e5T^{2} \)
83 \( 1 + 955.T + 5.71e5T^{2} \)
89 \( 1 - 617.T + 7.04e5T^{2} \)
97 \( 1 + 428. iT - 9.12e5T^{2} \)
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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

−18.48617913487145927155436321325, −17.14924552711350070885031856146, −16.01545679256366392263980675776, −13.64459895664640960911155622437, −12.69245070233277704580363555412, −11.10281217338752786649014441118, −8.716488189713775148320336662018, −8.325899958509834730509700289978, −6.16330868745957486881641457251, −1.22109800368859220164441475038, 4.20523770250516741168475357012, 7.24207365458814842960906234474, 9.164635252183473952165056881961, 10.30434893899211389086971043941, 10.94014436599962510429839631532, 13.79662102442458929136121135343, 15.28466929073193466234551466208, 16.33929830630615198291593715847, 17.54344921328433851587204445272, 18.55503630037313831858307576295

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