↑

Properties

Label 2-17e2-17.2-c1-0-13
Degree $2$
Conductor $289$
Sign $-0.961 + 0.275i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s − 0.999i·4-s + (−0.765 − 1.84i)5-s + (1.53 − 3.69i)7-s + (−2.12 + 2.12i)8-s + (−2.12 + 2.12i)9-s + (−0.765 + 1.84i)10-s + 2i·13-s + (−3.69 + 1.53i)14-s + 1.00·16-s + 3·18-s + (−2.82 − 2.82i)19-s + (−1.84 + 0.765i)20-s + (−3.69 − 1.53i)23-s + (0.707 − 0.707i)25-s + (1.41 − 1.41i)26-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s − 0.499i·4-s + (−0.342 − 0.826i)5-s + (0.578 − 1.39i)7-s + (−0.750 + 0.750i)8-s + (−0.707 + 0.707i)9-s + (−0.242 + 0.584i)10-s + 0.554i·13-s + (−0.987 + 0.409i)14-s + 0.250·16-s + 0.707·18-s + (−0.648 − 0.648i)19-s + (−0.413 + 0.171i)20-s + (−0.770 − 0.319i)23-s + (0.141 − 0.141i)25-s + (0.277 − 0.277i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.961 + 0.275i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.961 + 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100374 - 0.713784i\)
\(L(\frac12)\) \(\approx\) \(0.100374 - 0.713784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T + 2iT^{2} \)
3 \( 1 + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.765 + 1.84i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.53 + 3.69i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 + (2.82 + 2.82i)T + 19iT^{2} \)
23 \( 1 + (3.69 + 1.53i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.29 + 5.54i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.69 + 1.53i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.84 + 0.765i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.29 - 5.54i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-2.82 + 2.82i)T - 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 + (-8.48 + 8.48i)T - 59iT^{2} \)
61 \( 1 + (-3.82 + 9.23i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-3.69 + 1.53i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.29 + 5.54i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-11.0 - 4.59i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-2.82 - 2.82i)T + 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (0.765 + 1.84i)T + (-68.5 + 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.19404335646009565083133915810, −10.57055621693463971554917827170, −9.593285546609606616389732600868, −8.505355987977238323301935903270, −7.87065804890524287231228230036, −6.42635013392396773270046341939, −5.04703290419720831473774142565, −4.21639436182380443339416007427, −2.16933983264778584539643925041, −0.62497419398261959914472369781, 2.62910998232784470428584633312, 3.66768610800993715881367741692, 5.54463440645621299841802258031, 6.42371734545121169417803174997, 7.53628591221710317050915731345, 8.483066282495300592345985042133, 8.981760305622564811521478854140, 10.28143263741546218469409409540, 11.52903958196578572842170272916, 12.01920615652872121804871393327

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