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Properties

Label 2-17-17.16-c7-0-0
Degree $2$
Conductor $17$
Sign $-0.881 + 0.471i$
Analytic cond. $5.31054$
Root an. cond. $2.30446$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57·2-s + 88.6i·3-s − 121.·4-s − 321. i·5-s − 228. i·6-s − 569. i·7-s + 641.·8-s − 5.66e3·9-s + 828. i·10-s + 4.67e3i·11-s − 1.07e4i·12-s − 7.57e3·13-s + 1.46e3i·14-s + 2.85e4·15-s + 1.38e4·16-s + (−1.78e4 + 9.55e3i)17-s + ⋯
L(s)  = 1  − 0.227·2-s + 1.89i·3-s − 0.948·4-s − 1.15i·5-s − 0.431i·6-s − 0.627i·7-s + 0.443·8-s − 2.59·9-s + 0.262i·10-s + 1.05i·11-s − 1.79i·12-s − 0.956·13-s + 0.142i·14-s + 2.18·15-s + 0.847·16-s + (−0.881 + 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(5.31054\)
Root analytic conductor: \(2.30446\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :7/2),\ -0.881 + 0.471i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0649153 - 0.258930i\)
\(L(\frac12)\) \(\approx\) \(0.0649153 - 0.258930i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.78e4 - 9.55e3i)T \)
good2 \( 1 + 2.57T + 128T^{2} \)
3 \( 1 - 88.6iT - 2.18e3T^{2} \)
5 \( 1 + 321. iT - 7.81e4T^{2} \)
7 \( 1 + 569. iT - 8.23e5T^{2} \)
11 \( 1 - 4.67e3iT - 1.94e7T^{2} \)
13 \( 1 + 7.57e3T + 6.27e7T^{2} \)
19 \( 1 + 3.17e4T + 8.93e8T^{2} \)
23 \( 1 + 9.40e3iT - 3.40e9T^{2} \)
29 \( 1 - 2.86e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.36e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.22e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.40e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.50e5T + 2.71e11T^{2} \)
47 \( 1 - 2.96e5T + 5.06e11T^{2} \)
53 \( 1 + 1.69e6T + 1.17e12T^{2} \)
59 \( 1 + 5.50e5T + 2.48e12T^{2} \)
61 \( 1 + 1.70e6iT - 3.14e12T^{2} \)
67 \( 1 - 3.41e5T + 6.06e12T^{2} \)
71 \( 1 + 1.13e6iT - 9.09e12T^{2} \)
73 \( 1 - 3.30e6iT - 1.10e13T^{2} \)
79 \( 1 - 2.00e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.55e6T + 2.71e13T^{2} \)
89 \( 1 - 5.45e4T + 4.42e13T^{2} \)
97 \( 1 + 7.11e6iT - 8.07e13T^{2} \)
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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

−17.24733376220566310883296183192, −17.01226934078969433638134885090, −15.51872081065884163937180157201, −14.33307038881247547514190003587, −12.64598198585030929812353852078, −10.51861312201993638455024333259, −9.534975299104574447175570195146, −8.546017331942757774789766732040, −4.87590344811477354480675188213, −4.29331355207883854402684811028, 0.17226911251707662695759496085, 2.51525513794626326949820602634, 6.04070305908466817144100765063, 7.47982335834818396942097798873, 8.834047128078352738338718948165, 11.13726614010925519113375748250, 12.59923920737767883332617744522, 13.72891421528669621318394061815, 14.64868266429978569125399720686, 17.22409764846973193707838701874

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