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Properties

Label 2-17-17.4-c5-0-5
Degree $2$
Conductor $17$
Sign $-0.853 - 0.521i$
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  − 10.4i·2-s + (−15.8 + 15.8i)3-s − 77.9·4-s + (28.0 − 28.0i)5-s + (165. + 165. i)6-s + (−131. − 131. i)7-s + 481. i·8-s − 258. i·9-s + (−293. − 293. i)10-s + (−220. − 220. i)11-s + (1.23e3 − 1.23e3i)12-s + 201.·13-s + (−1.37e3 + 1.37e3i)14-s + 886. i·15-s + 2.55e3·16-s + (1.11e3 − 419. i)17-s + ⋯
L(s)  = 1  − 1.85i·2-s + (−1.01 + 1.01i)3-s − 2.43·4-s + (0.501 − 0.501i)5-s + (1.88 + 1.88i)6-s + (−1.01 − 1.01i)7-s + 2.65i·8-s − 1.06i·9-s + (−0.928 − 0.928i)10-s + (−0.549 − 0.549i)11-s + (2.47 − 2.47i)12-s + 0.331·13-s + (−1.87 + 1.87i)14-s + 1.01i·15-s + 2.49·16-s + (0.935 − 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.853 - 0.521i$
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: \((2,\ 17,\ (\ :5/2),\ -0.853 - 0.521i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.136976 + 0.487310i\)
\(L(\frac12)\) \(\approx\) \(0.136976 + 0.487310i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.11e3 + 419. i)T \)
good2 \( 1 + 10.4iT - 32T^{2} \)
3 \( 1 + (15.8 - 15.8i)T - 243iT^{2} \)
5 \( 1 + (-28.0 + 28.0i)T - 3.12e3iT^{2} \)
7 \( 1 + (131. + 131. i)T + 1.68e4iT^{2} \)
11 \( 1 + (220. + 220. i)T + 1.61e5iT^{2} \)
13 \( 1 - 201.T + 3.71e5T^{2} \)
19 \( 1 + 1.10e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.01e3 + 1.01e3i)T + 6.43e6iT^{2} \)
29 \( 1 + (3.59e3 - 3.59e3i)T - 2.05e7iT^{2} \)
31 \( 1 + (1.04e3 - 1.04e3i)T - 2.86e7iT^{2} \)
37 \( 1 + (-5.87e3 + 5.87e3i)T - 6.93e7iT^{2} \)
41 \( 1 + (2.91e3 + 2.91e3i)T + 1.15e8iT^{2} \)
43 \( 1 + 1.31e4iT - 1.47e8T^{2} \)
47 \( 1 - 4.18e3T + 2.29e8T^{2} \)
53 \( 1 + 2.37e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.89e4iT - 7.14e8T^{2} \)
61 \( 1 + (3.96e4 + 3.96e4i)T + 8.44e8iT^{2} \)
67 \( 1 - 6.84e4T + 1.35e9T^{2} \)
71 \( 1 + (4.55e3 - 4.55e3i)T - 1.80e9iT^{2} \)
73 \( 1 + (3.53e3 - 3.53e3i)T - 2.07e9iT^{2} \)
79 \( 1 + (-1.95e3 - 1.95e3i)T + 3.07e9iT^{2} \)
83 \( 1 - 5.38e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.87e4T + 5.58e9T^{2} \)
97 \( 1 + (-9.11e4 + 9.11e4i)T - 8.58e9iT^{2} \)
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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

−17.17763628039929337021571116880, −16.26231746732889065528668401799, −13.70331362336270982931327613542, −12.67745062850671139370992632808, −11.13852279361314499369194235434, −10.29535000496867502492745793966, −9.359471742856686108897413439402, −5.28498378571159034198569154671, −3.65477344853896193810017064448, −0.46118701197008085810983169629, 5.75209778511565423584407848163, 6.30756330345847298362380717488, 7.75462139871484319717314140951, 9.733734649718593547682618650396, 12.36485691253558670563536827499, 13.38764868573510797291414854811, 14.93149551931493568327796117970, 16.16250714047341990346483041846, 17.23301154305049583850985573639, 18.37213972443132409033326300087

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