Properties

Label 2-17-17.13-c3-0-0
Degree $2$
Conductor $17$
Sign $0.0794 - 0.996i$
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  + 3.93i·2-s + (−0.299 − 0.299i)3-s − 7.49·4-s + (1.37 + 1.37i)5-s + (1.18 − 1.18i)6-s + (17.9 − 17.9i)7-s + 1.98i·8-s − 26.8i·9-s + (−5.43 + 5.43i)10-s + (−22.9 + 22.9i)11-s + (2.24 + 2.24i)12-s − 54.6·13-s + (70.7 + 70.7i)14-s − 0.827i·15-s − 67.7·16-s + (58.5 − 38.6i)17-s + ⋯
L(s)  = 1  + 1.39i·2-s + (−0.0576 − 0.0576i)3-s − 0.937·4-s + (0.123 + 0.123i)5-s + (0.0803 − 0.0803i)6-s + (0.971 − 0.971i)7-s + 0.0876i·8-s − 0.993i·9-s + (−0.171 + 0.171i)10-s + (−0.628 + 0.628i)11-s + (0.0540 + 0.0540i)12-s − 1.16·13-s + (1.35 + 1.35i)14-s − 0.0142i·15-s − 1.05·16-s + (0.834 − 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0794 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0794 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.0794 - 0.996i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.0794 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.770448 + 0.711480i\)
\(L(\frac12)\) \(\approx\) \(0.770448 + 0.711480i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-58.5 + 38.6i)T \)
good2 \( 1 - 3.93iT - 8T^{2} \)
3 \( 1 + (0.299 + 0.299i)T + 27iT^{2} \)
5 \( 1 + (-1.37 - 1.37i)T + 125iT^{2} \)
7 \( 1 + (-17.9 + 17.9i)T - 343iT^{2} \)
11 \( 1 + (22.9 - 22.9i)T - 1.33e3iT^{2} \)
13 \( 1 + 54.6T + 2.19e3T^{2} \)
19 \( 1 - 46.1iT - 6.85e3T^{2} \)
23 \( 1 + (53.2 - 53.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (-111. - 111. i)T + 2.43e4iT^{2} \)
31 \( 1 + (-178. - 178. i)T + 2.97e4iT^{2} \)
37 \( 1 + (159. + 159. i)T + 5.06e4iT^{2} \)
41 \( 1 + (163. - 163. i)T - 6.89e4iT^{2} \)
43 \( 1 - 119. iT - 7.95e4T^{2} \)
47 \( 1 + 188.T + 1.03e5T^{2} \)
53 \( 1 + 468. iT - 1.48e5T^{2} \)
59 \( 1 - 751. iT - 2.05e5T^{2} \)
61 \( 1 + (-341. + 341. i)T - 2.26e5iT^{2} \)
67 \( 1 - 533.T + 3.00e5T^{2} \)
71 \( 1 + (-55.2 - 55.2i)T + 3.57e5iT^{2} \)
73 \( 1 + (-270. - 270. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-904. + 904. i)T - 4.93e5iT^{2} \)
83 \( 1 + 591. iT - 5.71e5T^{2} \)
89 \( 1 + 609.T + 7.04e5T^{2} \)
97 \( 1 + (-1.12e3 - 1.12e3i)T + 9.12e5iT^{2} \)
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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

−17.97013751046614985204991060575, −17.46963618939547821983031156446, −16.15388149957201961788129509186, −14.76267946005561731468930433121, −14.09921733224927811427658374916, −12.06218860639229549429044897538, −10.06657264973037884645170986044, −8.008786003122701774754681558182, −6.90608271651863634775251382964, −4.96089457649031419736760280302, 2.35290567019488074780878588608, 5.07866348333236228723257195587, 8.202857578659883471533014340052, 10.01424881619890158473548202141, 11.26796509089315284975336018579, 12.32486694535558400135049577736, 13.74796874259667595204760022263, 15.41606224451845123523889432713, 17.14184133733697507747555522381, 18.66566312093127738390460586079

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