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Properties

Label 2-17-17.13-c3-0-2
Degree $2$
Conductor $17$
Sign $0.0774 + 0.996i$
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  − 2.11i·2-s + (−5.92 − 5.92i)3-s + 3.51·4-s + (10.1 + 10.1i)5-s + (−12.5 + 12.5i)6-s + (3.21 − 3.21i)7-s − 24.3i·8-s + 43.2i·9-s + (21.5 − 21.5i)10-s + (−34.0 + 34.0i)11-s + (−20.8 − 20.8i)12-s + 49.5·13-s + (−6.81 − 6.81i)14-s − 120. i·15-s − 23.5·16-s + (−51.7 + 47.2i)17-s + ⋯
L(s)  = 1  − 0.748i·2-s + (−1.14 − 1.14i)3-s + 0.439·4-s + (0.908 + 0.908i)5-s + (−0.854 + 0.854i)6-s + (0.173 − 0.173i)7-s − 1.07i·8-s + 1.60i·9-s + (0.680 − 0.680i)10-s + (−0.932 + 0.932i)11-s + (−0.501 − 0.501i)12-s + 1.05·13-s + (−0.130 − 0.130i)14-s − 2.07i·15-s − 0.367·16-s + (−0.738 + 0.674i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0774 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0774 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.0774 + 0.996i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.0774 + 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.694016 - 0.642174i\)
\(L(\frac12)\) \(\approx\) \(0.694016 - 0.642174i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (51.7 - 47.2i)T \)
good2 \( 1 + 2.11iT - 8T^{2} \)
3 \( 1 + (5.92 + 5.92i)T + 27iT^{2} \)
5 \( 1 + (-10.1 - 10.1i)T + 125iT^{2} \)
7 \( 1 + (-3.21 + 3.21i)T - 343iT^{2} \)
11 \( 1 + (34.0 - 34.0i)T - 1.33e3iT^{2} \)
13 \( 1 - 49.5T + 2.19e3T^{2} \)
19 \( 1 + 31.1iT - 6.85e3T^{2} \)
23 \( 1 + (29.6 - 29.6i)T - 1.21e4iT^{2} \)
29 \( 1 + (85.8 + 85.8i)T + 2.43e4iT^{2} \)
31 \( 1 + (-118. - 118. i)T + 2.97e4iT^{2} \)
37 \( 1 + (251. + 251. i)T + 5.06e4iT^{2} \)
41 \( 1 + (44.8 - 44.8i)T - 6.89e4iT^{2} \)
43 \( 1 + 87.7iT - 7.95e4T^{2} \)
47 \( 1 - 281.T + 1.03e5T^{2} \)
53 \( 1 - 5.63iT - 1.48e5T^{2} \)
59 \( 1 + 134. iT - 2.05e5T^{2} \)
61 \( 1 + (-422. + 422. i)T - 2.26e5iT^{2} \)
67 \( 1 + 951.T + 3.00e5T^{2} \)
71 \( 1 + (-234. - 234. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-18.7 - 18.7i)T + 3.89e5iT^{2} \)
79 \( 1 + (-23.4 + 23.4i)T - 4.93e5iT^{2} \)
83 \( 1 + 283. iT - 5.71e5T^{2} \)
89 \( 1 + 191.T + 7.04e5T^{2} \)
97 \( 1 + (-203. - 203. i)T + 9.12e5iT^{2} \)
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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

−18.11988383328987441765240898040, −17.46289190234193742368571323895, −15.63080035433143018971910980274, −13.54984542822915281150781709307, −12.54010450581521140228673519320, −11.13123943954672311569063603067, −10.38096540505646707706333085275, −7.16939616777459365669093710526, −6.07904238286820764077112287950, −1.98982379708749078810849432379, 5.17228505607946442845161025837, 6.04820426604958289261007283157, 8.697789739397150969722747861959, 10.45319363093597240771569218293, 11.58188250467845456860471421130, 13.54510045452236554457464851304, 15.47174309755473959275287449120, 16.25053065938806319033198045159, 16.95360438603420992478493933909, 18.11205083576912316842013528135

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