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Properties

Label 2-17-17.13-c5-0-3
Degree $2$
Conductor $17$
Sign $0.925 + 0.379i$
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  − 4.80i·2-s + (15.6 + 15.6i)3-s + 8.88·4-s + (−8.57 − 8.57i)5-s + (75.4 − 75.4i)6-s + (38.9 − 38.9i)7-s − 196. i·8-s + 249. i·9-s + (−41.2 + 41.2i)10-s + (−492. + 492. i)11-s + (139. + 139. i)12-s − 46.9·13-s + (−187. − 187. i)14-s − 269. i·15-s − 660.·16-s + (−322. − 1.14e3i)17-s + ⋯
L(s)  = 1  − 0.849i·2-s + (1.00 + 1.00i)3-s + 0.277·4-s + (−0.153 − 0.153i)5-s + (0.855 − 0.855i)6-s + (0.300 − 0.300i)7-s − 1.08i·8-s + 1.02i·9-s + (−0.130 + 0.130i)10-s + (−1.22 + 1.22i)11-s + (0.279 + 0.279i)12-s − 0.0769·13-s + (−0.255 − 0.255i)14-s − 0.308i·15-s − 0.645·16-s + (−0.270 − 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.925 + 0.379i$
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: \((2,\ 17,\ (\ :5/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.80694 - 0.356128i\)
\(L(\frac12)\) \(\approx\) \(1.80694 - 0.356128i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (322. + 1.14e3i)T \)
good2 \( 1 + 4.80iT - 32T^{2} \)
3 \( 1 + (-15.6 - 15.6i)T + 243iT^{2} \)
5 \( 1 + (8.57 + 8.57i)T + 3.12e3iT^{2} \)
7 \( 1 + (-38.9 + 38.9i)T - 1.68e4iT^{2} \)
11 \( 1 + (492. - 492. i)T - 1.61e5iT^{2} \)
13 \( 1 + 46.9T + 3.71e5T^{2} \)
19 \( 1 - 67.9iT - 2.47e6T^{2} \)
23 \( 1 + (2.88e3 - 2.88e3i)T - 6.43e6iT^{2} \)
29 \( 1 + (1.17e3 + 1.17e3i)T + 2.05e7iT^{2} \)
31 \( 1 + (-4.29e3 - 4.29e3i)T + 2.86e7iT^{2} \)
37 \( 1 + (-9.24e3 - 9.24e3i)T + 6.93e7iT^{2} \)
41 \( 1 + (-1.13e4 + 1.13e4i)T - 1.15e8iT^{2} \)
43 \( 1 + 5.38e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.11e4T + 2.29e8T^{2} \)
53 \( 1 + 668. iT - 4.18e8T^{2} \)
59 \( 1 - 968. iT - 7.14e8T^{2} \)
61 \( 1 + (2.98e4 - 2.98e4i)T - 8.44e8iT^{2} \)
67 \( 1 - 3.02e4T + 1.35e9T^{2} \)
71 \( 1 + (2.89e4 + 2.89e4i)T + 1.80e9iT^{2} \)
73 \( 1 + (1.59e4 + 1.59e4i)T + 2.07e9iT^{2} \)
79 \( 1 + (-5.74e4 + 5.74e4i)T - 3.07e9iT^{2} \)
83 \( 1 - 9.30e4iT - 3.93e9T^{2} \)
89 \( 1 + 6.58e4T + 5.58e9T^{2} \)
97 \( 1 + (2.76e4 + 2.76e4i)T + 8.58e9iT^{2} \)
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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

−18.02821472633082066770517909888, −15.97034019294847861025126671756, −15.33061811394927137523492177529, −13.74833610195704806584343448714, −12.12700165001089540734520301797, −10.48671570675292851268017886630, −9.601147489427474755573933829793, −7.67953888106476374264324423439, −4.37220786278705442996707630291, −2.57077029616211471097516814052, 2.48204040015554833814961603671, 6.04082808423382702210780113063, 7.72515763645983710806981432142, 8.414092201144085398385064393417, 11.07026544593375508968845084608, 12.89250400003481573300202937505, 14.13299066679542754717824517504, 15.16312337252772180577102438443, 16.43311222303049477840147887026, 18.08072826508487793455535968672

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