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Properties

Label 2-17e2-17.16-c1-0-11
Degree $2$
Conductor $289$
Sign $0.970 + 0.242i$
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  + 2.30·2-s − 1.30i·3-s + 3.30·4-s + 2.30i·5-s − 3i·6-s + 0.302i·7-s + 3.00·8-s + 1.30·9-s + 5.30i·10-s − 3i·11-s − 4.30i·12-s − 3.30·13-s + 0.697i·14-s + 3·15-s + 0.302·16-s + ⋯
L(s)  = 1  + 1.62·2-s − 0.752i·3-s + 1.65·4-s + 1.02i·5-s − 1.22i·6-s + 0.114i·7-s + 1.06·8-s + 0.434·9-s + 1.67i·10-s − 0.904i·11-s − 1.24i·12-s − 0.916·13-s + 0.186i·14-s + 0.774·15-s + 0.0756·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88993 - 0.355767i\)
\(L(\frac12)\) \(\approx\) \(2.88993 - 0.355767i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 + 1.30iT - 3T^{2} \)
5 \( 1 - 2.30iT - 5T^{2} \)
7 \( 1 - 0.302iT - 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
19 \( 1 + 5.90T + 19T^{2} \)
23 \( 1 + 2.30iT - 23T^{2} \)
29 \( 1 - 9.90iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 0.605iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 2.09T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 8.81iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 3.21iT - 71T^{2} \)
73 \( 1 - 0.394iT - 73T^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 - 3.21T + 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−12.27903010789664160269508586010, −11.04251530854045900155310452697, −10.44235836520265008517664041909, −8.726494561065662511562972439206, −7.23547644733373997605511936723, −6.74459287197148895437943889181, −5.81041078732646165503022680908, −4.54963428732879217212389565932, −3.27906260025063364377979722862, −2.25504994659742440502227641625, 2.26350086311836236810123052886, 4.05526647325682379495121925185, 4.51025531549126923008860426578, 5.32276770502506408914670982726, 6.57684939579304217081878569939, 7.76919037603583222095851528179, 9.245615835640813122374148953986, 10.01829437691983885928955998000, 11.19265604426638155145452652228, 12.27109791008562083092715463582

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