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Properties

Label 2-17-17.9-c1-0-0
Degree $2$
Conductor $17$
Sign $0.997 - 0.0758i$
Analytic cond. $0.135745$
Root an. cond. $0.368436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 + 0.292i)2-s + (−2.41 − i)3-s + 1.82i·4-s + (0.707 − 1.70i)5-s + (1 − 0.414i)6-s + (0.414 + i)7-s + (−1.12 − 1.12i)8-s + (2.70 + 2.70i)9-s + (0.292 + 0.707i)10-s + (−1 + 0.414i)11-s + (1.82 − 4.41i)12-s − 1.41i·13-s + (−0.414 − 0.171i)14-s + (−3.41 + 3.41i)15-s − 3·16-s + (−2.82 + 3i)17-s + ⋯
L(s)  = 1  + (−0.207 + 0.207i)2-s + (−1.39 − 0.577i)3-s + 0.914i·4-s + (0.316 − 0.763i)5-s + (0.408 − 0.169i)6-s + (0.156 + 0.377i)7-s + (−0.396 − 0.396i)8-s + (0.902 + 0.902i)9-s + (0.0926 + 0.223i)10-s + (−0.301 + 0.124i)11-s + (0.527 − 1.27i)12-s − 0.392i·13-s + (−0.110 − 0.0458i)14-s + (−0.881 + 0.881i)15-s − 0.750·16-s + (−0.685 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(0.135745\)
Root analytic conductor: \(0.368436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :1/2),\ 0.997 - 0.0758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405101 + 0.0153929i\)
\(L(\frac12)\) \(\approx\) \(0.405101 + 0.0153929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.82 - 3i)T \)
good2 \( 1 + (0.292 - 0.292i)T - 2iT^{2} \)
3 \( 1 + (2.41 + i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.707 + 1.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.414 - i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (1 - 0.414i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
19 \( 1 + (-3.41 + 3.41i)T - 19iT^{2} \)
23 \( 1 + (-3.82 + 1.58i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.70 - 4.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (3 + 1.24i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.53 + 1.46i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.12 + 7.53i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.41 + 3.41i)T + 43iT^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (4.24 + 4.24i)T + 59iT^{2} \)
61 \( 1 + (-3.53 - 8.53i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 6.82T + 67T^{2} \)
71 \( 1 + (-12.0 - 5i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.05 - 4.94i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.82 - 1.58i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.242 - 0.242i)T - 83iT^{2} \)
89 \( 1 + 9.41iT - 89T^{2} \)
97 \( 1 + (-2.46 + 5.94i)T + (-68.5 - 68.5i)T^{2} \)
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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

−18.48794488140891694503217332982, −17.52958343992373978457674666457, −16.90789875753421801561708669843, −15.69138481605947645870435591367, −13.08640749870537083344509537076, −12.41733962896031951736224781186, −11.10378052582036485546872080009, −8.833155096843014552661580438105, −7.04494793081621256455477703736, −5.27609823497030340629124026891, 5.15470058723758459368169597079, 6.59514495921055644029448722027, 9.734396513385092223719084388957, 10.74244975396732498786477290357, 11.57899485898614895591759569503, 13.87946730539711479922404677570, 15.27820992234551772161971224693, 16.56524358333855849691711895352, 17.88951454681442042745088118000, 18.67069722775521702538317120898

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