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Properties

Label 2-17-17.4-c5-0-4
Degree $2$
Conductor $17$
Sign $-0.492 + 0.870i$
Analytic cond. $2.72652$
Root an. cond. $1.65121$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.30i·2-s + (11.9 − 11.9i)3-s − 21.3·4-s + (−21.7 + 21.7i)5-s + (−87.6 − 87.6i)6-s + (6.10 + 6.10i)7-s − 77.7i·8-s − 44.6i·9-s + (158. + 158. i)10-s + (224. + 224. i)11-s + (−256. + 256. i)12-s + 914.·13-s + (44.6 − 44.6i)14-s + 522. i·15-s − 1.25e3·16-s + (−456. − 1.10e3i)17-s + ⋯
L(s)  = 1  − 1.29i·2-s + (0.769 − 0.769i)3-s − 0.667·4-s + (−0.389 + 0.389i)5-s + (−0.993 − 0.993i)6-s + (0.0471 + 0.0471i)7-s − 0.429i·8-s − 0.183i·9-s + (0.502 + 0.502i)10-s + (0.559 + 0.559i)11-s + (−0.513 + 0.513i)12-s + 1.50·13-s + (0.0608 − 0.0608i)14-s + 0.599i·15-s − 1.22·16-s + (−0.383 − 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(2.72652\)
Root analytic conductor: \(1.65121\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :5/2),\ -0.492 + 0.870i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.823529 - 1.41189i\)
\(L(\frac12)\) \(\approx\) \(0.823529 - 1.41189i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (456. + 1.10e3i)T \)
good2 \( 1 + 7.30iT - 32T^{2} \)
3 \( 1 + (-11.9 + 11.9i)T - 243iT^{2} \)
5 \( 1 + (21.7 - 21.7i)T - 3.12e3iT^{2} \)
7 \( 1 + (-6.10 - 6.10i)T + 1.68e4iT^{2} \)
11 \( 1 + (-224. - 224. i)T + 1.61e5iT^{2} \)
13 \( 1 - 914.T + 3.71e5T^{2} \)
19 \( 1 - 1.51e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.61e3 + 1.61e3i)T + 6.43e6iT^{2} \)
29 \( 1 + (238. - 238. i)T - 2.05e7iT^{2} \)
31 \( 1 + (7.38e3 - 7.38e3i)T - 2.86e7iT^{2} \)
37 \( 1 + (467. - 467. i)T - 6.93e7iT^{2} \)
41 \( 1 + (4.43e3 + 4.43e3i)T + 1.15e8iT^{2} \)
43 \( 1 + 1.27e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.15e3T + 2.29e8T^{2} \)
53 \( 1 - 2.00e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.31e4iT - 7.14e8T^{2} \)
61 \( 1 + (-2.33e4 - 2.33e4i)T + 8.44e8iT^{2} \)
67 \( 1 + 2.00e4T + 1.35e9T^{2} \)
71 \( 1 + (5.91e3 - 5.91e3i)T - 1.80e9iT^{2} \)
73 \( 1 + (2.37e4 - 2.37e4i)T - 2.07e9iT^{2} \)
79 \( 1 + (8.69e3 + 8.69e3i)T + 3.07e9iT^{2} \)
83 \( 1 + 1.15e5iT - 3.93e9T^{2} \)
89 \( 1 - 6.60e4T + 5.58e9T^{2} \)
97 \( 1 + (-7.83e4 + 7.83e4i)T - 8.58e9iT^{2} \)
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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

−18.27849937869908578983122600334, −16.00501331853660493367643321477, −14.29076852680384319955532365174, −13.12286790388500413211705254952, −11.89986220271234716820836234763, −10.62469725451670618720788290629, −8.831235966294194749280700249456, −7.06055985836394594020013822819, −3.53470609511760951264613293488, −1.73598354479876631465904138511, 4.02553078239592579720319098897, 6.18033137041197914467274884055, 8.176852859207882439404406284884, 9.072506089823426518883320259027, 11.23601154429267228638014559218, 13.50743114247159416270728713057, 14.75378072289798004540998071376, 15.67294499843096139410735394482, 16.46684694921747392164332986850, 17.89797752787534439726220669238

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