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Properties

Label 2-17e2-17.13-c1-0-3
Degree $2$
Conductor $289$
Sign $-0.816 - 0.577i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.87i·2-s + (1.79 + 1.79i)3-s − 1.53·4-s + (−0.0852 − 0.0852i)5-s + (−3.36 + 3.36i)6-s + (1.08 − 1.08i)7-s + 0.879i·8-s + 3.41i·9-s + (0.160 − 0.160i)10-s + (1.90 − 1.90i)11-s + (−2.74 − 2.74i)12-s − 4.57·13-s + (2.03 + 2.03i)14-s − 0.305i·15-s − 4.71·16-s + ⋯
L(s)  = 1  + 1.32i·2-s + (1.03 + 1.03i)3-s − 0.766·4-s + (−0.0381 − 0.0381i)5-s + (−1.37 + 1.37i)6-s + (0.409 − 0.409i)7-s + 0.310i·8-s + 1.13i·9-s + (0.0506 − 0.0506i)10-s + (0.574 − 0.574i)11-s + (−0.791 − 0.791i)12-s − 1.26·13-s + (0.544 + 0.544i)14-s − 0.0788i·15-s − 1.17·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545321 + 1.71401i\)
\(L(\frac12)\) \(\approx\) \(0.545321 + 1.71401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 - 1.87iT - 2T^{2} \)
3 \( 1 + (-1.79 - 1.79i)T + 3iT^{2} \)
5 \( 1 + (0.0852 + 0.0852i)T + 5iT^{2} \)
7 \( 1 + (-1.08 + 1.08i)T - 7iT^{2} \)
11 \( 1 + (-1.90 + 1.90i)T - 11iT^{2} \)
13 \( 1 + 4.57T + 13T^{2} \)
19 \( 1 - 1.87iT - 19T^{2} \)
23 \( 1 + (-5.24 + 5.24i)T - 23iT^{2} \)
29 \( 1 + (2.41 + 2.41i)T + 29iT^{2} \)
31 \( 1 + (-2.71 - 2.71i)T + 31iT^{2} \)
37 \( 1 + (3.71 + 3.71i)T + 37iT^{2} \)
41 \( 1 + (4.41 - 4.41i)T - 41iT^{2} \)
43 \( 1 - 5.49iT - 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 + 0.822iT - 53T^{2} \)
59 \( 1 + 3.14iT - 59T^{2} \)
61 \( 1 + (0.867 - 0.867i)T - 61iT^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 + (0.120 + 0.120i)T + 71iT^{2} \)
73 \( 1 + (-8.34 - 8.34i)T + 73iT^{2} \)
79 \( 1 + (2.93 - 2.93i)T - 79iT^{2} \)
83 \( 1 - 2.43iT - 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + (7.91 + 7.91i)T + 97iT^{2} \)
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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

−12.24916273756145391734161860066, −10.99225952783415266308984947116, −10.01321243749578463225611839001, −9.015109260241244069482416102596, −8.349808010331342948071754486220, −7.47919067881430786906428677026, −6.39846517154029840105256555557, −4.99897922144694147374557498190, −4.22751431817057861659436001936, −2.70966310375258598665062181181, 1.52925943102606852433895366958, 2.42365233184229769773480142945, 3.47600654104420789703165135452, 5.01165927435810782798649756511, 6.95625556985612729933640957612, 7.48518814755579025674525615859, 8.911761367583865000643724701630, 9.414752781584305359222527383710, 10.60303766483941002920092914753, 11.80116622144683054149522755331

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