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Properties

Label 2-17-17.15-c5-0-4
Degree $2$
Conductor $17$
Sign $0.118 + 0.992i$
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  + (2.46 + 2.46i)2-s + (−10.5 − 25.5i)3-s − 19.8i·4-s + (−30.5 + 12.6i)5-s + (36.9 − 89.2i)6-s + (151. + 62.8i)7-s + (127. − 127. i)8-s + (−369. + 369. i)9-s + (−106. − 44.1i)10-s + (208. − 504. i)11-s + (−505. + 209. i)12-s + 359. i·13-s + (219. + 529. i)14-s + (646. + 646. i)15-s − 1.84·16-s + (1.16e3 − 231. i)17-s + ⋯
L(s)  = 1  + (0.436 + 0.436i)2-s + (−0.678 − 1.63i)3-s − 0.618i·4-s + (−0.545 + 0.226i)5-s + (0.419 − 1.01i)6-s + (1.17 + 0.484i)7-s + (0.706 − 0.706i)8-s + (−1.51 + 1.51i)9-s + (−0.337 − 0.139i)10-s + (0.520 − 1.25i)11-s + (−1.01 + 0.420i)12-s + 0.590i·13-s + (0.299 + 0.722i)14-s + (0.741 + 0.741i)15-s − 0.00180·16-s + (0.981 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(2.72652\)
Root analytic conductor: \(1.65121\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :5/2),\ 0.118 + 0.992i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.01221 - 0.898164i\)
\(L(\frac12)\) \(\approx\) \(1.01221 - 0.898164i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.16e3 + 231. i)T \)
good2 \( 1 + (-2.46 - 2.46i)T + 32iT^{2} \)
3 \( 1 + (10.5 + 25.5i)T + (-171. + 171. i)T^{2} \)
5 \( 1 + (30.5 - 12.6i)T + (2.20e3 - 2.20e3i)T^{2} \)
7 \( 1 + (-151. - 62.8i)T + (1.18e4 + 1.18e4i)T^{2} \)
11 \( 1 + (-208. + 504. i)T + (-1.13e5 - 1.13e5i)T^{2} \)
13 \( 1 - 359. iT - 3.71e5T^{2} \)
19 \( 1 + (92.6 + 92.6i)T + 2.47e6iT^{2} \)
23 \( 1 + (23.8 - 57.5i)T + (-4.55e6 - 4.55e6i)T^{2} \)
29 \( 1 + (-3.01e3 + 1.24e3i)T + (1.45e7 - 1.45e7i)T^{2} \)
31 \( 1 + (-3.54e3 - 8.54e3i)T + (-2.02e7 + 2.02e7i)T^{2} \)
37 \( 1 + (1.57e3 + 3.80e3i)T + (-4.90e7 + 4.90e7i)T^{2} \)
41 \( 1 + (-1.72e3 - 713. i)T + (8.19e7 + 8.19e7i)T^{2} \)
43 \( 1 + (8.77e3 - 8.77e3i)T - 1.47e8iT^{2} \)
47 \( 1 + 3.15e3iT - 2.29e8T^{2} \)
53 \( 1 + (1.15e4 + 1.15e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (1.61e4 - 1.61e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-4.48e4 - 1.85e4i)T + (5.97e8 + 5.97e8i)T^{2} \)
67 \( 1 - 4.53e4T + 1.35e9T^{2} \)
71 \( 1 + (-2.06e3 - 4.98e3i)T + (-1.27e9 + 1.27e9i)T^{2} \)
73 \( 1 + (2.68e4 - 1.11e4i)T + (1.46e9 - 1.46e9i)T^{2} \)
79 \( 1 + (-1.89e4 + 4.58e4i)T + (-2.17e9 - 2.17e9i)T^{2} \)
83 \( 1 + (-7.04e3 - 7.04e3i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.24e5iT - 5.58e9T^{2} \)
97 \( 1 + (9.77e4 - 4.04e4i)T + (6.07e9 - 6.07e9i)T^{2} \)
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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

−17.75154071126393490249896126000, −16.29297483380368881762798976229, −14.52141589963509781200713786570, −13.71668075284996884722315470063, −11.99090255325292145922547672410, −11.15896570462353207903437706060, −8.157493750406343535352519128267, −6.72098619883753327215971373540, −5.41844922647677435422924384803, −1.24802179901165505552922602969, 3.96950919484681477447745700740, 4.87709621520478309867807732809, 8.034390036040125476514238769196, 10.05361233104599057535876710179, 11.33069006129415645022461357943, 12.20721648472062841051460190904, 14.41435130154987816852663583793, 15.54745112072069326934569445065, 16.94421200371992619458621314089, 17.50793214043392534598904419367

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