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Properties

Label 2-17e2-17.16-c3-0-35
Degree $2$
Conductor $289$
Sign $0.911 + 0.410i$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.07·2-s + 5.02i·3-s − 6.85·4-s − 4.32i·5-s + 5.37i·6-s + 4.44i·7-s − 15.9·8-s + 1.79·9-s − 4.63i·10-s − 68.9i·11-s − 34.4i·12-s − 27.1·13-s + 4.76i·14-s + 21.7·15-s + 37.7·16-s + ⋯
L(s)  = 1  + 0.378·2-s + 0.966i·3-s − 0.856·4-s − 0.386i·5-s + 0.365i·6-s + 0.240i·7-s − 0.703·8-s + 0.0666·9-s − 0.146i·10-s − 1.89i·11-s − 0.827i·12-s − 0.580·13-s + 0.0909i·14-s + 0.373·15-s + 0.590·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.911 + 0.410i$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.585405090\)
\(L(\frac12)\) \(\approx\) \(1.585405090\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.07T + 8T^{2} \)
3 \( 1 - 5.02iT - 27T^{2} \)
5 \( 1 + 4.32iT - 125T^{2} \)
7 \( 1 - 4.44iT - 343T^{2} \)
11 \( 1 + 68.9iT - 1.33e3T^{2} \)
13 \( 1 + 27.1T + 2.19e3T^{2} \)
19 \( 1 - 123.T + 6.85e3T^{2} \)
23 \( 1 - 98.6iT - 1.21e4T^{2} \)
29 \( 1 + 21.9iT - 2.43e4T^{2} \)
31 \( 1 + 241. iT - 2.97e4T^{2} \)
37 \( 1 + 324. iT - 5.06e4T^{2} \)
41 \( 1 + 164. iT - 6.89e4T^{2} \)
43 \( 1 + 383.T + 7.95e4T^{2} \)
47 \( 1 - 411.T + 1.03e5T^{2} \)
53 \( 1 - 380.T + 1.48e5T^{2} \)
59 \( 1 + 63.3T + 2.05e5T^{2} \)
61 \( 1 + 37.1iT - 2.26e5T^{2} \)
67 \( 1 + 48.2T + 3.00e5T^{2} \)
71 \( 1 + 672. iT - 3.57e5T^{2} \)
73 \( 1 + 562. iT - 3.89e5T^{2} \)
79 \( 1 + 461. iT - 4.93e5T^{2} \)
83 \( 1 + 972.T + 5.71e5T^{2} \)
89 \( 1 - 16.0T + 7.04e5T^{2} \)
97 \( 1 + 598. iT - 9.12e5T^{2} \)
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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

−11.32017438165775517651372567175, −10.24190757085599847603673667610, −9.281199203591351252888277075617, −8.830240101379265242142174587288, −7.57927205366104421750991358399, −5.73419430175139184236565820594, −5.22626851639109529483359844170, −4.02904224586953816416407431600, −3.18535003775333058826930189182, −0.64883891576342669161435625119, 1.21333160184238665891572656100, 2.78204647357398764274604023455, 4.35373483768234017140294074135, 5.19820185079685396259625305336, 6.82431257575763116223162222794, 7.24490965203241414069995110827, 8.429666869625149872951139279093, 9.719290848310765161719883375445, 10.24528068787647486733631030370, 11.97856776942953134791017467724

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