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Properties

Label 2-17e2-17.15-c1-0-10
Degree $2$
Conductor $289$
Sign $0.932 + 0.361i$
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  + (1.32 + 1.32i)2-s + (−0.968 − 2.33i)3-s + 1.53i·4-s + (−0.111 + 0.0461i)5-s + (1.82 − 4.39i)6-s + (1.41 + 0.586i)7-s + (0.621 − 0.621i)8-s + (−2.41 + 2.41i)9-s + (−0.209 − 0.0867i)10-s + (1.03 − 2.48i)11-s + (3.58 − 1.48i)12-s − 4.57i·13-s + (1.10 + 2.66i)14-s + (0.215 + 0.215i)15-s + 4.71·16-s + ⋯
L(s)  = 1  + (0.939 + 0.939i)2-s + (−0.559 − 1.35i)3-s + 0.766i·4-s + (−0.0498 + 0.0206i)5-s + (0.743 − 1.79i)6-s + (0.534 + 0.221i)7-s + (0.219 − 0.219i)8-s + (−0.804 + 0.804i)9-s + (−0.0662 − 0.0274i)10-s + (0.310 − 0.750i)11-s + (1.03 − 0.428i)12-s − 1.26i·13-s + (0.294 + 0.710i)14-s + (0.0557 + 0.0557i)15-s + 1.17·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.932 + 0.361i$
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: \((2,\ 289,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80211 - 0.337451i\)
\(L(\frac12)\) \(\approx\) \(1.80211 - 0.337451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 1.32i)T + 2iT^{2} \)
3 \( 1 + (0.968 + 2.33i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.111 - 0.0461i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.41 - 0.586i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.03 + 2.48i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 4.57iT - 13T^{2} \)
19 \( 1 + (-1.32 - 1.32i)T + 19iT^{2} \)
23 \( 1 + (2.83 - 6.84i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-3.15 + 1.30i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.46 - 3.54i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.00 - 4.84i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.77 + 2.39i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.88 - 3.88i)T - 43iT^{2} \)
47 \( 1 - 7.34iT - 47T^{2} \)
53 \( 1 + (0.581 + 0.581i)T + 53iT^{2} \)
59 \( 1 + (2.22 - 2.22i)T - 59iT^{2} \)
61 \( 1 + (-1.13 - 0.469i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 + (-0.0651 - 0.157i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-10.9 + 4.51i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.58 - 3.83i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.72 + 1.72i)T + 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + (-10.3 + 4.28i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.99178296363414211316977359069, −11.28489778774658639988871367367, −9.947890548127619376697743577743, −8.155599882340125754572540241123, −7.69295111558301974706875749679, −6.61062570235104878838408921263, −5.81839845447804014382431323803, −5.15421396341688542551963738393, −3.44719523126652311152711638486, −1.33407145073756168461680595595, 2.15581195158830225481025003402, 3.89961077778027428350330274312, 4.42733386448151063844428891822, 5.17864770637603587059759739101, 6.61218550862801267093041465907, 8.200708876356822403854955168800, 9.537648290074050080275978015920, 10.26126086451306676064929460882, 11.10321320877975285549531044974, 11.75244047366751578752044739418

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