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Properties

Label 2-17-17.13-c5-0-5
Degree $2$
Conductor $17$
Sign $-0.965 + 0.260i$
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  − 8.15i·2-s + (−6.93 − 6.93i)3-s − 34.5·4-s + (−2.80 − 2.80i)5-s + (−56.5 + 56.5i)6-s + (−29.5 + 29.5i)7-s + 20.9i·8-s − 146. i·9-s + (−22.8 + 22.8i)10-s + (342. − 342. i)11-s + (239. + 239. i)12-s + 349.·13-s + (240. + 240. i)14-s + 38.8i·15-s − 935.·16-s + (952. + 715. i)17-s + ⋯
L(s)  = 1  − 1.44i·2-s + (−0.444 − 0.444i)3-s − 1.08·4-s + (−0.0501 − 0.0501i)5-s + (−0.641 + 0.641i)6-s + (−0.227 + 0.227i)7-s + 0.115i·8-s − 0.604i·9-s + (−0.0723 + 0.0723i)10-s + (0.853 − 0.853i)11-s + (0.480 + 0.480i)12-s + 0.573·13-s + (0.328 + 0.328i)14-s + 0.0446i·15-s − 0.913·16-s + (0.799 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.965 + 0.260i$
Analytic conductor: \(2.72652\)
Root analytic conductor: \(1.65121\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :5/2),\ -0.965 + 0.260i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.145645 - 1.09784i\)
\(L(\frac12)\) \(\approx\) \(0.145645 - 1.09784i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-952. - 715. i)T \)
good2 \( 1 + 8.15iT - 32T^{2} \)
3 \( 1 + (6.93 + 6.93i)T + 243iT^{2} \)
5 \( 1 + (2.80 + 2.80i)T + 3.12e3iT^{2} \)
7 \( 1 + (29.5 - 29.5i)T - 1.68e4iT^{2} \)
11 \( 1 + (-342. + 342. i)T - 1.61e5iT^{2} \)
13 \( 1 - 349.T + 3.71e5T^{2} \)
19 \( 1 - 475. iT - 2.47e6T^{2} \)
23 \( 1 + (431. - 431. i)T - 6.43e6iT^{2} \)
29 \( 1 + (-5.78e3 - 5.78e3i)T + 2.05e7iT^{2} \)
31 \( 1 + (-2.11e3 - 2.11e3i)T + 2.86e7iT^{2} \)
37 \( 1 + (2.23e3 + 2.23e3i)T + 6.93e7iT^{2} \)
41 \( 1 + (-1.20e4 + 1.20e4i)T - 1.15e8iT^{2} \)
43 \( 1 + 1.26e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.48e4T + 2.29e8T^{2} \)
53 \( 1 - 4.52e3iT - 4.18e8T^{2} \)
59 \( 1 - 7.86e3iT - 7.14e8T^{2} \)
61 \( 1 + (3.13e3 - 3.13e3i)T - 8.44e8iT^{2} \)
67 \( 1 + 4.58e4T + 1.35e9T^{2} \)
71 \( 1 + (-1.59e4 - 1.59e4i)T + 1.80e9iT^{2} \)
73 \( 1 + (-2.74e4 - 2.74e4i)T + 2.07e9iT^{2} \)
79 \( 1 + (-3.10e4 + 3.10e4i)T - 3.07e9iT^{2} \)
83 \( 1 + 8.09e3iT - 3.93e9T^{2} \)
89 \( 1 - 1.27e5T + 5.58e9T^{2} \)
97 \( 1 + (-2.78e4 - 2.78e4i)T + 8.58e9iT^{2} \)
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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

−17.73749041694686619357968998131, −16.13000136890509069977358923591, −14.12023973785415710797720273186, −12.55263238414480539587083618853, −11.85301796003053713751915208444, −10.51321060325083494796758167073, −8.912842262475001724549182599854, −6.31504031380521649172619896211, −3.53850445306177487808557132007, −1.06584355037100205324100340402, 4.72903298672008554060646381997, 6.37235245994047194314144049721, 7.85467821383751904919692777339, 9.696471751812573715234933354750, 11.50397972966269429102032931163, 13.56006721628730942250533451732, 14.83847019379496863753170075896, 16.00584432036208916385840276725, 16.81772056630365074345930222045, 17.83868189445840882351365755978

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