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Properties

Label 2-17-17.4-c15-0-0
Degree $2$
Conductor $17$
Sign $0.348 - 0.937i$
Analytic cond. $24.2578$
Root an. cond. $4.92523$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 250. i·2-s + (−4.19e3 + 4.19e3i)3-s − 3.00e4·4-s + (1.14e5 − 1.14e5i)5-s + (1.05e6 + 1.05e6i)6-s + (−2.34e6 − 2.34e6i)7-s − 6.78e5i·8-s − 2.07e7i·9-s + (−2.87e7 − 2.87e7i)10-s + (2.13e6 + 2.13e6i)11-s + (1.25e8 − 1.25e8i)12-s + 1.18e8·13-s + (−5.88e8 + 5.88e8i)14-s + 9.62e8i·15-s − 1.15e9·16-s + (−8.87e8 − 1.44e9i)17-s + ⋯
L(s)  = 1  − 1.38i·2-s + (−1.10 + 1.10i)3-s − 0.917·4-s + (0.657 − 0.657i)5-s + (1.53 + 1.53i)6-s + (−1.07 − 1.07i)7-s − 0.114i·8-s − 1.44i·9-s + (−0.910 − 0.910i)10-s + (0.0330 + 0.0330i)11-s + (1.01 − 1.01i)12-s + 0.524·13-s + (−1.49 + 1.49i)14-s + 1.45i·15-s − 1.07·16-s + (−0.524 − 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(24.2578\)
Root analytic conductor: \(4.92523\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :15/2),\ 0.348 - 0.937i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.08360030940\)
\(L(\frac12)\) \(\approx\) \(0.08360030940\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (8.87e8 + 1.44e9i)T \)
good2 \( 1 + 250. iT - 3.27e4T^{2} \)
3 \( 1 + (4.19e3 - 4.19e3i)T - 1.43e7iT^{2} \)
5 \( 1 + (-1.14e5 + 1.14e5i)T - 3.05e10iT^{2} \)
7 \( 1 + (2.34e6 + 2.34e6i)T + 4.74e12iT^{2} \)
11 \( 1 + (-2.13e6 - 2.13e6i)T + 4.17e15iT^{2} \)
13 \( 1 - 1.18e8T + 5.11e16T^{2} \)
19 \( 1 - 5.77e9iT - 1.51e19T^{2} \)
23 \( 1 + (1.65e9 + 1.65e9i)T + 2.66e20iT^{2} \)
29 \( 1 + (-8.46e10 + 8.46e10i)T - 8.62e21iT^{2} \)
31 \( 1 + (1.46e11 - 1.46e11i)T - 2.34e22iT^{2} \)
37 \( 1 + (3.49e11 - 3.49e11i)T - 3.33e23iT^{2} \)
41 \( 1 + (-1.39e12 - 1.39e12i)T + 1.55e24iT^{2} \)
43 \( 1 - 1.85e12iT - 3.17e24T^{2} \)
47 \( 1 + 3.73e12T + 1.20e25T^{2} \)
53 \( 1 - 1.93e11iT - 7.31e25T^{2} \)
59 \( 1 + 4.09e12iT - 3.65e26T^{2} \)
61 \( 1 + (9.12e12 + 9.12e12i)T + 6.02e26iT^{2} \)
67 \( 1 + 7.10e13T + 2.46e27T^{2} \)
71 \( 1 + (-6.88e13 + 6.88e13i)T - 5.87e27iT^{2} \)
73 \( 1 + (-7.23e11 + 7.23e11i)T - 8.90e27iT^{2} \)
79 \( 1 + (-2.02e14 - 2.02e14i)T + 2.91e28iT^{2} \)
83 \( 1 + 3.17e14iT - 6.11e28T^{2} \)
89 \( 1 + 5.40e14T + 1.74e29T^{2} \)
97 \( 1 + (-1.27e14 + 1.27e14i)T - 6.33e29iT^{2} \)
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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

−16.05755385396314996849697537412, −13.60188878853250418006528720005, −12.47817173694718268580743165313, −11.14218092939156898689604321916, −10.12862354243960129848908804018, −9.508878220627560912595410935074, −6.29632841464499152497266608460, −4.61794402631020703117206748270, −3.45293914768054886223416377429, −1.19220163049059536078071497332, 0.03528925691670841412894076557, 2.27794157807403390650002513422, 5.57523491725005378531438916972, 6.30382765842344120933094745063, 7.01612019597125893468672526042, 8.906538445638906592689907951656, 11.00571131955502871375576476840, 12.55908935344958997226072776296, 13.65765872110147874338279831789, 15.26361287959942559872342278833

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