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Properties

Label 2-17-17.9-c3-0-0
Degree $2$
Conductor $17$
Sign $-0.995 - 0.0948i$
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  + (−3.43 + 3.43i)2-s + (−4.67 − 1.93i)3-s − 15.6i·4-s + (−7.10 + 17.1i)5-s + (22.7 − 9.42i)6-s + (5.36 + 12.9i)7-s + (26.2 + 26.2i)8-s + (−0.947 − 0.947i)9-s + (−34.5 − 83.3i)10-s + (17.9 − 7.45i)11-s + (−30.3 + 73.1i)12-s + 29.9i·13-s + (−62.9 − 26.0i)14-s + (66.4 − 66.4i)15-s − 55.3·16-s + (−56.0 + 42.0i)17-s + ⋯
L(s)  = 1  + (−1.21 + 1.21i)2-s + (−0.900 − 0.373i)3-s − 1.95i·4-s + (−0.635 + 1.53i)5-s + (1.54 − 0.641i)6-s + (0.289 + 0.699i)7-s + (1.16 + 1.16i)8-s + (−0.0351 − 0.0351i)9-s + (−1.09 − 2.63i)10-s + (0.493 − 0.204i)11-s + (−0.729 + 1.76i)12-s + 0.638i·13-s + (−1.20 − 0.497i)14-s + (1.14 − 1.14i)15-s − 0.865·16-s + (−0.799 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.995 - 0.0948i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ -0.995 - 0.0948i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0148496 + 0.312561i\)
\(L(\frac12)\) \(\approx\) \(0.0148496 + 0.312561i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (56.0 - 42.0i)T \)
good2 \( 1 + (3.43 - 3.43i)T - 8iT^{2} \)
3 \( 1 + (4.67 + 1.93i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (7.10 - 17.1i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-5.36 - 12.9i)T + (-242. + 242. i)T^{2} \)
11 \( 1 + (-17.9 + 7.45i)T + (941. - 941. i)T^{2} \)
13 \( 1 - 29.9iT - 2.19e3T^{2} \)
19 \( 1 + (32.3 - 32.3i)T - 6.85e3iT^{2} \)
23 \( 1 + (-82.5 + 34.1i)T + (8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (22.1 - 53.4i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (-149. - 61.9i)T + (2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-125. - 51.8i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (21.7 + 52.5i)T + (-4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (-36.8 - 36.8i)T + 7.95e4iT^{2} \)
47 \( 1 + 482. iT - 1.03e5T^{2} \)
53 \( 1 + (374. - 374. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-198. - 198. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-224. - 541. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + 367.T + 3.00e5T^{2} \)
71 \( 1 + (55.9 + 23.1i)T + (2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (140. - 340. i)T + (-2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-201. + 83.3i)T + (3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (-420. + 420. i)T - 5.71e5iT^{2} \)
89 \( 1 - 887. iT - 7.04e5T^{2} \)
97 \( 1 + (-338. + 817. i)T + (-6.45e5 - 6.45e5i)T^{2} \)
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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

−18.59691976547196762809196852643, −17.82456982057596418220177745799, −16.74542136674755946075444086051, −15.30549253690496707914896775559, −14.57257538644409241157615270854, −11.74165843101822075734412610748, −10.64521313628523086143978856417, −8.691271109992251204613585417169, −6.99999897075463800161405171289, −6.17943458880210868926969019175, 0.62705736002593820342591753138, 4.58067247302031778353835872755, 8.043221214254806507621515305500, 9.368252058913560505083447970565, 10.92794322600932929766563074865, 11.76284708908072927821761031687, 12.99996095452768517722541859834, 15.95395498561106678233050896691, 17.10254536661075833323153092653, 17.49436237669021546589138219570

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