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Properties

Label 2-17-17.13-c9-0-9
Degree $2$
Conductor $17$
Sign $-0.993 + 0.109i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.4i·2-s + (22.6 + 22.6i)3-s + 52.9·4-s + (−1.05e3 − 1.05e3i)5-s + (485. − 485. i)6-s + (−862. + 862. i)7-s − 1.21e4i·8-s − 1.86e4i·9-s + (−2.26e4 + 2.26e4i)10-s + (−4.23e4 + 4.23e4i)11-s + (1.20e3 + 1.20e3i)12-s − 1.43e5·13-s + (1.84e4 + 1.84e4i)14-s − 4.79e4i·15-s − 2.32e5·16-s + (2.40e5 + 2.46e5i)17-s + ⋯
L(s)  = 1  − 0.946i·2-s + (0.161 + 0.161i)3-s + 0.103·4-s + (−0.755 − 0.755i)5-s + (0.153 − 0.153i)6-s + (−0.135 + 0.135i)7-s − 1.04i·8-s − 0.947i·9-s + (−0.715 + 0.715i)10-s + (−0.872 + 0.872i)11-s + (0.0167 + 0.0167i)12-s − 1.38·13-s + (0.128 + 0.128i)14-s − 0.244i·15-s − 0.885·16-s + (0.697 + 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.993 + 0.109i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0650161 - 1.18436i\)
\(L(\frac12)\) \(\approx\) \(0.0650161 - 1.18436i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.40e5 - 2.46e5i)T \)
good2 \( 1 + 21.4iT - 512T^{2} \)
3 \( 1 + (-22.6 - 22.6i)T + 1.96e4iT^{2} \)
5 \( 1 + (1.05e3 + 1.05e3i)T + 1.95e6iT^{2} \)
7 \( 1 + (862. - 862. i)T - 4.03e7iT^{2} \)
11 \( 1 + (4.23e4 - 4.23e4i)T - 2.35e9iT^{2} \)
13 \( 1 + 1.43e5T + 1.06e10T^{2} \)
19 \( 1 + 8.97e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.17e6 + 1.17e6i)T - 1.80e12iT^{2} \)
29 \( 1 + (-2.86e6 - 2.86e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-3.45e5 - 3.45e5i)T + 2.64e13iT^{2} \)
37 \( 1 + (6.84e6 + 6.84e6i)T + 1.29e14iT^{2} \)
41 \( 1 + (-1.19e7 + 1.19e7i)T - 3.27e14iT^{2} \)
43 \( 1 - 6.52e6iT - 5.02e14T^{2} \)
47 \( 1 - 5.90e7T + 1.11e15T^{2} \)
53 \( 1 - 1.27e7iT - 3.29e15T^{2} \)
59 \( 1 + 4.63e7iT - 8.66e15T^{2} \)
61 \( 1 + (-2.97e7 + 2.97e7i)T - 1.16e16iT^{2} \)
67 \( 1 + 2.39e8T + 2.72e16T^{2} \)
71 \( 1 + (-3.70e7 - 3.70e7i)T + 4.58e16iT^{2} \)
73 \( 1 + (8.98e7 + 8.98e7i)T + 5.88e16iT^{2} \)
79 \( 1 + (-8.21e7 + 8.21e7i)T - 1.19e17iT^{2} \)
83 \( 1 + 3.57e8iT - 1.86e17T^{2} \)
89 \( 1 - 4.24e8T + 3.50e17T^{2} \)
97 \( 1 + (-5.85e8 - 5.85e8i)T + 7.60e17iT^{2} \)
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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

−15.90507745943057745211693690147, −14.98812960795525658495662915841, −12.56989187788290115880779173575, −12.25721385969146566658929948165, −10.51433358856707956651462957307, −9.147645517571175660734416973924, −7.20252145565112312081027030636, −4.55258126871898072703867375486, −2.73824929212544878336236957927, −0.55481093257018729746387879974, 2.79522396965431755808325375825, 5.40288479048271892942137402700, 7.30464243164603729562728501842, 7.953838079291691814231136092656, 10.45140032274486536669647397799, 11.79617306071323902623740007741, 13.79927277921995265729105655351, 14.88760882930086131583620089774, 16.02342445068511107739141670559, 16.95005230233801377115286764698

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