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Properties

Label 2-17e2-17.10-c2-0-5
Degree $2$
Conductor $289$
Sign $-0.528 - 0.848i$
Analytic cond. $7.87467$
Root an. cond. $2.80618$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s + (−3.89 − 2.60i)3-s + (2.12 + 2.12i)4-s + (4.60 − 0.915i)5-s + (3.89 − 2.60i)6-s + (−9.20 − 1.83i)7-s + (−6.46 + 2.67i)8-s + (4.97 + 12.0i)9-s + (−0.915 + 4.60i)10-s + (2.60 + 3.89i)11-s + (−2.74 − 13.8i)12-s + (7.07 − 7.07i)13-s + (5.21 − 7.79i)14-s + (−20.3 − 8.41i)15-s + 5.00i·16-s + ⋯
L(s)  = 1  + (−0.191 + 0.461i)2-s + (−1.29 − 0.868i)3-s + (0.530 + 0.530i)4-s + (0.920 − 0.183i)5-s + (0.649 − 0.434i)6-s + (−1.31 − 0.261i)7-s + (−0.808 + 0.334i)8-s + (0.552 + 1.33i)9-s + (−0.0915 + 0.460i)10-s + (0.236 + 0.354i)11-s + (−0.228 − 1.15i)12-s + (0.543 − 0.543i)13-s + (0.372 − 0.557i)14-s + (−1.35 − 0.561i)15-s + 0.312i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.528 - 0.848i$
Analytic conductor: \(7.87467\)
Root analytic conductor: \(2.80618\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1),\ -0.528 - 0.848i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.294999 + 0.531397i\)
\(L(\frac12)\) \(\approx\) \(0.294999 + 0.531397i\)
\(L(2)\) 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 \( 1 + (0.382 - 0.923i)T + (-2.82 - 2.82i)T^{2} \)
3 \( 1 + (3.89 + 2.60i)T + (3.44 + 8.31i)T^{2} \)
5 \( 1 + (-4.60 + 0.915i)T + (23.0 - 9.56i)T^{2} \)
7 \( 1 + (9.20 + 1.83i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (-2.60 - 3.89i)T + (-46.3 + 111. i)T^{2} \)
13 \( 1 + (-7.07 + 7.07i)T - 169iT^{2} \)
19 \( 1 + (6.88 - 16.6i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (7.79 - 5.21i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (-10.0 - 50.6i)T + (-776. + 321. i)T^{2} \)
31 \( 1 + (20.8 - 31.1i)T + (-367. - 887. i)T^{2} \)
37 \( 1 + (3.89 + 2.60i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (9.20 + 1.83i)T + (1.55e3 + 643. i)T^{2} \)
43 \( 1 + (-14.5 - 35.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (41.0 - 41.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (6.88 - 16.6i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-68.3 + 28.3i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (4.57 - 23.0i)T + (-3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 + 34iT - 4.48e3T^{2} \)
71 \( 1 + (70.1 + 46.9i)T + (1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (-82.8 + 16.4i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (41.6 + 62.3i)T + (-2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (-53.5 - 22.1i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (55.1 + 55.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (-25.6 - 128. i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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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

−12.17267328015367703187177078871, −11.00432200228354313402583338843, −10.16623821797342075712951005057, −9.015968487030411774031122048866, −7.67714623397328716896120294642, −6.68578554716683976821758750805, −6.25570101360569127202676912412, −5.42605781872940648948115194778, −3.34197465307809387025784239304, −1.58442936996435373688810109722, 0.35914405222309323813317444074, 2.35931423392607325930287083144, 3.91138116070654602610068244502, 5.51026264780321191239312648354, 6.19844447851453581874089286976, 6.64526088542844494871960791453, 9.022961454886474870503235371521, 9.877545366995644126006254492009, 10.16364284273205743248891686440, 11.27022928820414186592089291420

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