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Properties

Label 2-17e2-17.13-c1-0-13
Degree $2$
Conductor $289$
Sign $-0.519 + 0.854i$
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  − 1.53i·2-s + (0.952 + 0.952i)3-s − 0.347·4-s + (−2.49 − 2.49i)5-s + (1.45 − 1.45i)6-s + (0.245 − 0.245i)7-s − 2.53i·8-s − 1.18i·9-s + (−3.82 + 3.82i)10-s + (−1.24 + 1.24i)11-s + (−0.330 − 0.330i)12-s + 3.29·13-s + (−0.376 − 0.376i)14-s − 4.75i·15-s − 4.57·16-s + ⋯
L(s)  = 1  − 1.08i·2-s + (0.550 + 0.550i)3-s − 0.173·4-s + (−1.11 − 1.11i)5-s + (0.595 − 0.595i)6-s + (0.0928 − 0.0928i)7-s − 0.895i·8-s − 0.394i·9-s + (−1.21 + 1.21i)10-s + (−0.374 + 0.374i)11-s + (−0.0955 − 0.0955i)12-s + 0.912·13-s + (−0.100 − 0.100i)14-s − 1.22i·15-s − 1.14·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.519 + 0.854i$
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: \((2,\ 289,\ (\ :1/2),\ -0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668483 - 1.18949i\)
\(L(\frac12)\) \(\approx\) \(0.668483 - 1.18949i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53iT - 2T^{2} \)
3 \( 1 + (-0.952 - 0.952i)T + 3iT^{2} \)
5 \( 1 + (2.49 + 2.49i)T + 5iT^{2} \)
7 \( 1 + (-0.245 + 0.245i)T - 7iT^{2} \)
11 \( 1 + (1.24 - 1.24i)T - 11iT^{2} \)
13 \( 1 - 3.29T + 13T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
23 \( 1 + (-1.99 + 1.99i)T - 23iT^{2} \)
29 \( 1 + (-0.837 - 0.837i)T + 29iT^{2} \)
31 \( 1 + (-5.02 - 5.02i)T + 31iT^{2} \)
37 \( 1 + (2.77 + 2.77i)T + 37iT^{2} \)
41 \( 1 + (3.47 - 3.47i)T - 41iT^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 - 8.36iT - 53T^{2} \)
59 \( 1 + 10.8iT - 59T^{2} \)
61 \( 1 + (-3.11 + 3.11i)T - 61iT^{2} \)
67 \( 1 - 8.07T + 67T^{2} \)
71 \( 1 + (-7.95 - 7.95i)T + 71iT^{2} \)
73 \( 1 + (1.20 + 1.20i)T + 73iT^{2} \)
79 \( 1 + (-8.18 + 8.18i)T - 79iT^{2} \)
83 \( 1 - 2.14iT - 83T^{2} \)
89 \( 1 + 6.41T + 89T^{2} \)
97 \( 1 + (2.88 + 2.88i)T + 97iT^{2} \)
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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

−11.48604754601175884875829409136, −10.70462757402531684771903171678, −9.628173373864470014256167108868, −8.824092018855140777370572913805, −8.010370740506512183092411567276, −6.64209588629981857911870209021, −4.76139958817556055207128149478, −3.97085407307155142355251521432, −2.99092552720099331095133245991, −1.04508298395846203181823094937, 2.43513568717294009873516260200, 3.67967342739965986100220467955, 5.37420054353761780371125009705, 6.60289094634250810830103086429, 7.28259726270525477285265018958, 8.081310755477526079777790730507, 8.572241168515139867408027181300, 10.45778911233917191008198195851, 11.19783346613599704509293201571, 11.98373097934626378081948754687

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