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Properties

Label 2-17e2-17.15-c1-0-6
Degree $2$
Conductor $289$
Sign $0.516 + 0.856i$
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  + (−1.08 − 1.08i)2-s + (0.515 + 1.24i)3-s + 0.347i·4-s + (3.26 − 1.35i)5-s + (0.789 − 1.90i)6-s + (−0.320 − 0.132i)7-s + (−1.79 + 1.79i)8-s + (0.837 − 0.837i)9-s + (−4.99 − 2.07i)10-s + (0.673 − 1.62i)11-s + (−0.432 + 0.179i)12-s + 3.29i·13-s + (0.203 + 0.491i)14-s + (3.36 + 3.36i)15-s + 4.57·16-s + ⋯
L(s)  = 1  + (−0.766 − 0.766i)2-s + (0.297 + 0.718i)3-s + 0.173i·4-s + (1.45 − 0.604i)5-s + (0.322 − 0.778i)6-s + (−0.121 − 0.0502i)7-s + (−0.633 + 0.633i)8-s + (0.279 − 0.279i)9-s + (−1.58 − 0.654i)10-s + (0.202 − 0.489i)11-s + (−0.124 + 0.0516i)12-s + 0.912i·13-s + (0.0544 + 0.131i)14-s + (0.868 + 0.868i)15-s + 1.14·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03714 - 0.585959i\)
\(L(\frac12)\) \(\approx\) \(1.03714 - 0.585959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 1.08i)T + 2iT^{2} \)
3 \( 1 + (-0.515 - 1.24i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.26 + 1.35i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.320 + 0.132i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.673 + 1.62i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.29iT - 13T^{2} \)
19 \( 1 + (1.08 + 1.08i)T + 19iT^{2} \)
23 \( 1 + (-1.07 + 2.60i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.09 + 0.453i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.71 + 6.56i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.50 + 3.62i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.54 - 1.88i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.76 - 7.76i)T - 43iT^{2} \)
47 \( 1 - 5.12iT - 47T^{2} \)
53 \( 1 + (-5.91 - 5.91i)T + 53iT^{2} \)
59 \( 1 + (7.68 - 7.68i)T - 59iT^{2} \)
61 \( 1 + (-4.07 - 1.68i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 + (-4.30 - 10.3i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-1.57 + 0.651i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.43 - 10.7i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.51 + 1.51i)T + 83iT^{2} \)
89 \( 1 - 6.41iT - 89T^{2} \)
97 \( 1 + (3.77 - 1.56i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.32226581554857001695482182956, −10.43777426079603044032082973784, −9.562855245554351115426938399760, −9.308862790364795572981746914551, −8.474317854564612386096028077552, −6.56167224959110963244640582956, −5.59518112587043093448084187640, −4.31562285449641295074801331056, −2.65213298299740890047046112533, −1.35185581216845951905476183582, 1.75522815099497320475163202918, 3.14102643054634227342371416607, 5.34038248183976074403358369810, 6.47168554785824387452409064879, 7.05685161502166800397786785737, 8.000565365398809754888510097599, 8.999995296793050633878665283986, 9.950785572231172037434085575986, 10.53997188749437917635748329677, 12.24203027172620931500887259813

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