Properties

Label 2-17-17.16-c9-0-8
Degree $2$
Conductor $17$
Sign $-0.282 + 0.959i$
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  + 11.8·2-s + 59.5i·3-s − 371.·4-s − 633. i·5-s + 706. i·6-s − 1.09e4i·7-s − 1.04e4·8-s + 1.61e4·9-s − 7.51e3i·10-s − 2.61e4i·11-s − 2.21e4i·12-s − 1.32e5·13-s − 1.29e5i·14-s + 3.77e4·15-s + 6.59e4·16-s + (9.71e4 − 3.30e5i)17-s + ⋯
L(s)  = 1  + 0.524·2-s + 0.424i·3-s − 0.725·4-s − 0.453i·5-s + 0.222i·6-s − 1.72i·7-s − 0.904·8-s + 0.819·9-s − 0.237i·10-s − 0.539i·11-s − 0.308i·12-s − 1.28·13-s − 0.901i·14-s + 0.192·15-s + 0.251·16-s + (0.282 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.282 + 0.959i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.756031 - 1.01027i\)
\(L(\frac12)\) \(\approx\) \(0.756031 - 1.01027i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-9.71e4 + 3.30e5i)T \)
good2 \( 1 - 11.8T + 512T^{2} \)
3 \( 1 - 59.5iT - 1.96e4T^{2} \)
5 \( 1 + 633. iT - 1.95e6T^{2} \)
7 \( 1 + 1.09e4iT - 4.03e7T^{2} \)
11 \( 1 + 2.61e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.32e5T + 1.06e10T^{2} \)
19 \( 1 + 8.34e5T + 3.22e11T^{2} \)
23 \( 1 - 1.27e6iT - 1.80e12T^{2} \)
29 \( 1 - 6.34e6iT - 1.45e13T^{2} \)
31 \( 1 + 8.52e6iT - 2.64e13T^{2} \)
37 \( 1 + 8.05e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.36e7iT - 3.27e14T^{2} \)
43 \( 1 - 2.89e6T + 5.02e14T^{2} \)
47 \( 1 - 2.63e7T + 1.11e15T^{2} \)
53 \( 1 - 2.45e6T + 3.29e15T^{2} \)
59 \( 1 + 8.84e6T + 8.66e15T^{2} \)
61 \( 1 - 6.18e7iT - 1.16e16T^{2} \)
67 \( 1 - 1.46e8T + 2.72e16T^{2} \)
71 \( 1 + 2.08e8iT - 4.58e16T^{2} \)
73 \( 1 + 7.44e7iT - 5.88e16T^{2} \)
79 \( 1 + 1.47e8iT - 1.19e17T^{2} \)
83 \( 1 + 5.69e7T + 1.86e17T^{2} \)
89 \( 1 - 1.03e9T + 3.50e17T^{2} \)
97 \( 1 - 1.03e9iT - 7.60e17T^{2} \)
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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

−16.46081526179338120424383821961, −14.75202965942492090447556824266, −13.62114682408936828709348347325, −12.60825713534052527642986849032, −10.55439008114787342679777528113, −9.313295266902893474655960318994, −7.32504024181043796152315935774, −4.91697991010355192109924487251, −3.87529523327844400879732534899, −0.54305372379285256143183400678, 2.40934031802655787283714915957, 4.68981573459531458337087955075, 6.37585127962757766144936945760, 8.445182933151621519665809281087, 9.955773684703541598658957072006, 12.29640301331616995717342609577, 12.73338729580187310061686912228, 14.64014097140272102378393747829, 15.24213376538413085229801626561, 17.40764716051361783465724455481

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