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Properties

Label 2-17-17.15-c3-0-1
Degree $2$
Conductor $17$
Sign $0.820 - 0.571i$
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  + (2.47 + 2.47i)2-s + (−1.08 − 2.61i)3-s + 4.24i·4-s + (−8.05 + 3.33i)5-s + (3.78 − 9.13i)6-s + (−6.33 − 2.62i)7-s + (9.28 − 9.28i)8-s + (13.4 − 13.4i)9-s + (−28.1 − 11.6i)10-s + (−23.6 + 57.0i)11-s + (11.0 − 4.59i)12-s + 5.37i·13-s + (−9.18 − 22.1i)14-s + (17.4 + 17.4i)15-s + 79.9·16-s + (44.2 + 54.3i)17-s + ⋯
L(s)  = 1  + (0.874 + 0.874i)2-s + (−0.208 − 0.502i)3-s + 0.530i·4-s + (−0.720 + 0.298i)5-s + (0.257 − 0.621i)6-s + (−0.341 − 0.141i)7-s + (0.410 − 0.410i)8-s + (0.498 − 0.498i)9-s + (−0.891 − 0.369i)10-s + (−0.648 + 1.56i)11-s + (0.266 − 0.110i)12-s + 0.114i·13-s + (−0.175 − 0.423i)14-s + (0.299 + 0.299i)15-s + 1.24·16-s + (0.631 + 0.774i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.820 - 0.571i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.26356 + 0.396509i\)
\(L(\frac12)\) \(\approx\) \(1.26356 + 0.396509i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-44.2 - 54.3i)T \)
good2 \( 1 + (-2.47 - 2.47i)T + 8iT^{2} \)
3 \( 1 + (1.08 + 2.61i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (8.05 - 3.33i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (6.33 + 2.62i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (23.6 - 57.0i)T + (-941. - 941. i)T^{2} \)
13 \( 1 - 5.37iT - 2.19e3T^{2} \)
19 \( 1 + (68.4 + 68.4i)T + 6.85e3iT^{2} \)
23 \( 1 + (-44.5 + 107. i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-182. + 75.5i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (52.8 + 127. i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-42.6 - 102. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (153. + 63.7i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (117. - 117. i)T - 7.95e4iT^{2} \)
47 \( 1 - 130. iT - 1.03e5T^{2} \)
53 \( 1 + (-505. - 505. i)T + 1.48e5iT^{2} \)
59 \( 1 + (598. - 598. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-4.61 - 1.91i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 314.T + 3.00e5T^{2} \)
71 \( 1 + (45.3 + 109. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (-601. + 248. i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (79.7 - 192. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (524. + 524. i)T + 5.71e5iT^{2} \)
89 \( 1 - 215. iT - 7.04e5T^{2} \)
97 \( 1 + (-651. + 270. i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−18.52530691114489712382240060176, −17.04836923787670282473325790038, −15.45119123394137398669954116800, −14.97704962007062671910417862113, −13.17636758318612798183850823732, −12.30265973607177801015530857952, −10.19548786802457507790978967606, −7.54642300684526191389181272974, −6.52557220776585370744090225359, −4.38480225524344062313562012776, 3.49237573874081310895319304065, 5.19602822545562950495250032332, 8.115061760395797536031792841677, 10.39218192938233042894863302454, 11.52027460876931611728831912530, 12.78180680395154390235282739332, 13.96225747456171289688488799653, 15.80516468628381750506445698075, 16.60410423637428696787008588599, 18.81566378647776441623504815352

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