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Properties

Label 2-17-17.9-c3-0-2
Degree $2$
Conductor $17$
Sign $0.746 + 0.665i$
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  + (2.22 − 2.22i)2-s + (−1.83 − 0.762i)3-s − 1.94i·4-s + (−1.91 + 4.62i)5-s + (−5.80 + 2.40i)6-s + (1.06 + 2.57i)7-s + (13.5 + 13.5i)8-s + (−16.2 − 16.2i)9-s + (6.04 + 14.5i)10-s + (−25.1 + 10.4i)11-s + (−1.47 + 3.57i)12-s − 59.7i·13-s + (8.11 + 3.36i)14-s + (7.05 − 7.05i)15-s + 75.7·16-s + (70.0 − 0.790i)17-s + ⋯
L(s)  = 1  + (0.788 − 0.788i)2-s + (−0.354 − 0.146i)3-s − 0.242i·4-s + (−0.171 + 0.413i)5-s + (−0.394 + 0.163i)6-s + (0.0575 + 0.138i)7-s + (0.596 + 0.596i)8-s + (−0.603 − 0.603i)9-s + (0.191 + 0.461i)10-s + (−0.689 + 0.285i)11-s + (−0.0355 + 0.0859i)12-s − 1.27i·13-s + (0.154 + 0.0641i)14-s + (0.121 − 0.121i)15-s + 1.18·16-s + (0.999 − 0.0112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.746 + 0.665i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20128 - 0.457375i\)
\(L(\frac12)\) \(\approx\) \(1.20128 - 0.457375i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-70.0 + 0.790i)T \)
good2 \( 1 + (-2.22 + 2.22i)T - 8iT^{2} \)
3 \( 1 + (1.83 + 0.762i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (1.91 - 4.62i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-1.06 - 2.57i)T + (-242. + 242. i)T^{2} \)
11 \( 1 + (25.1 - 10.4i)T + (941. - 941. i)T^{2} \)
13 \( 1 + 59.7iT - 2.19e3T^{2} \)
19 \( 1 + (-23.5 + 23.5i)T - 6.85e3iT^{2} \)
23 \( 1 + (194. - 80.7i)T + (8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-7.67 + 18.5i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (-123. - 51.1i)T + (2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-141. - 58.4i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (100. + 241. i)T + (-4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (224. + 224. i)T + 7.95e4iT^{2} \)
47 \( 1 - 329. iT - 1.03e5T^{2} \)
53 \( 1 + (219. - 219. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-38.7 - 38.7i)T + 2.05e5iT^{2} \)
61 \( 1 + (313. + 756. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + 731.T + 3.00e5T^{2} \)
71 \( 1 + (-581. - 240. i)T + (2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (189. - 458. i)T + (-2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-83.1 + 34.4i)T + (3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (257. - 257. i)T - 5.71e5iT^{2} \)
89 \( 1 - 192. iT - 7.04e5T^{2} \)
97 \( 1 + (-516. + 1.24e3i)T + (-6.45e5 - 6.45e5i)T^{2} \)
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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

−18.31407672525728572699326406147, −17.20851908058689660332698231623, −15.40821886289656431463093495059, −14.03783715951160659567550228254, −12.61957155695915369091982697148, −11.67956941135488594805263782240, −10.33274802455025213020382053074, −7.86632578310231062624988511328, −5.50055939819920528126720465314, −3.18412007922491424740425045963, 4.62884197489022035717104144423, 6.07273228724659865446232937980, 8.002828821243005778886454333504, 10.24603249627244519951376683375, 11.96296532257099424455524573791, 13.60731216615582276159608912654, 14.52013570412367007245665055242, 16.25389409081170567016829057354, 16.54562015820955345571297735625, 18.54431866019366594724207960837

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