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Properties

Label 2-17-17.15-c3-0-2
Degree $2$
Conductor $17$
Sign $-0.168 + 0.985i$
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  + (−1.89 − 1.89i)2-s + (−1.65 − 4.00i)3-s − 0.785i·4-s + (1.92 − 0.798i)5-s + (−4.45 + 10.7i)6-s + (23.0 + 9.56i)7-s + (−16.6 + 16.6i)8-s + (5.82 − 5.82i)9-s + (−5.17 − 2.14i)10-s + (1.46 − 3.53i)11-s + (−3.14 + 1.30i)12-s − 17.6i·13-s + (−25.6 − 62.0i)14-s + (−6.39 − 6.39i)15-s + 57.1·16-s + (−69.7 − 6.56i)17-s + ⋯
L(s)  = 1  + (−0.671 − 0.671i)2-s + (−0.318 − 0.770i)3-s − 0.0981i·4-s + (0.172 − 0.0714i)5-s + (−0.302 + 0.731i)6-s + (1.24 + 0.516i)7-s + (−0.737 + 0.737i)8-s + (0.215 − 0.215i)9-s + (−0.163 − 0.0678i)10-s + (0.0400 − 0.0967i)11-s + (−0.0755 + 0.0313i)12-s − 0.377i·13-s + (−0.490 − 1.18i)14-s + (−0.110 − 0.110i)15-s + 0.892·16-s + (−0.995 − 0.0936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.168 + 0.985i$
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: \((2,\ 17,\ (\ :3/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.488271 - 0.579079i\)
\(L(\frac12)\) \(\approx\) \(0.488271 - 0.579079i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (69.7 + 6.56i)T \)
good2 \( 1 + (1.89 + 1.89i)T + 8iT^{2} \)
3 \( 1 + (1.65 + 4.00i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.92 + 0.798i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-23.0 - 9.56i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (-1.46 + 3.53i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + 17.6iT - 2.19e3T^{2} \)
19 \( 1 + (-113. - 113. i)T + 6.85e3iT^{2} \)
23 \( 1 + (38.2 - 92.3i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-185. + 76.7i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (29.2 + 70.7i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (93.6 + 226. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (49.9 + 20.7i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (100. - 100. i)T - 7.95e4iT^{2} \)
47 \( 1 - 468. iT - 1.03e5T^{2} \)
53 \( 1 + (-68.2 - 68.2i)T + 1.48e5iT^{2} \)
59 \( 1 + (-257. + 257. i)T - 2.05e5iT^{2} \)
61 \( 1 + (653. + 270. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 304.T + 3.00e5T^{2} \)
71 \( 1 + (179. + 434. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (131. - 54.3i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (274. - 663. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (259. + 259. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (834. - 345. i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−17.99472959202485828409577744244, −17.68214782362393010922238380848, −15.41785944802904424129201645332, −13.97844528647923070044415736670, −12.15566274407536063484853998700, −11.26225403223893185039689323767, −9.590112461698514492186126839092, −7.944568844622349708167664172558, −5.69818131037012740520433512186, −1.59437853974255174968076439210, 4.64922345250640003947191320486, 7.07980552522260115628854611252, 8.614404935755556169642858024676, 10.20161420185664233510869875628, 11.62776076656994167876941055852, 13.72751698579107709479718390690, 15.34432056155478090216943919024, 16.30463761353391358799153429152, 17.46234998172401837433042212047, 18.17905572598260409807690557721

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