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Properties

Label 2-17e2-17.10-c2-0-12
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

\(L(\frac{3}{2})\) \(\approx\) \(1.15467 + 2.07997i\)
\(L(\frac12)\) \(\approx\) \(1.15467 + 2.07997i\)
\(L(2)\) 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

−11.55171522167536356178226566678, −11.13766686274539135700441571470, −9.851378294693060778748972808021, −8.587611546762746281719691687933, −8.020135447604422714238982240769, −7.75242149012461866742048884214, −5.98839143005321993289400652766, −4.40520365999435432196262797483, −3.50671699653232637427805468789, −2.36866328921802779450906136052, 1.20784995935468514650220715087, 2.19030460959938972430179822838, 3.52988584791887980245632101915, 4.95187400735785616541440440698, 6.80687340793320245574004556729, 7.46172293180429193797805621348, 8.448327621518551281403917863356, 9.039554456483593059053659012102, 10.51282746801442576922269180275, 11.34794810363281494711402823220

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