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Properties

Label 2-290-5.4-c1-0-11
Degree $2$
Conductor $290$
Sign $-0.683 + 0.729i$
Analytic cond. $2.31566$
Root an. cond. $1.52172$
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  + i·2-s − 2.81i·3-s − 4-s + (−1.63 − 1.52i)5-s + 2.81·6-s + 0.647i·7-s − i·8-s − 4.91·9-s + (1.52 − 1.63i)10-s − 3.05·11-s + 2.81i·12-s + 0.606i·13-s − 0.647·14-s + (−4.29 + 4.59i)15-s + 16-s − 2.27i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.62i·3-s − 0.5·4-s + (−0.729 − 0.683i)5-s + 1.14·6-s + 0.244i·7-s − 0.353i·8-s − 1.63·9-s + (0.483 − 0.516i)10-s − 0.921·11-s + 0.811i·12-s + 0.168i·13-s − 0.173·14-s + (−1.11 + 1.18i)15-s + 0.250·16-s − 0.551i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $-0.683 + 0.729i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ -0.683 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280506 - 0.647104i\)
\(L(\frac12)\) \(\approx\) \(0.280506 - 0.647104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.63 + 1.52i)T \)
29 \( 1 + T \)
good3 \( 1 + 2.81iT - 3T^{2} \)
7 \( 1 - 0.647iT - 7T^{2} \)
11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 - 0.606iT - 13T^{2} \)
17 \( 1 + 2.27iT - 17T^{2} \)
19 \( 1 + 3.91T + 19T^{2} \)
23 \( 1 + 8.36iT - 23T^{2} \)
31 \( 1 - 4.40T + 31T^{2} \)
37 \( 1 + 4.92iT - 37T^{2} \)
41 \( 1 - 9.74T + 41T^{2} \)
43 \( 1 + 1.30iT - 43T^{2} \)
47 \( 1 + 5.18iT - 47T^{2} \)
53 \( 1 - 2.11iT - 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 6.62T + 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 1.68T + 79T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 1.58T + 89T^{2} \)
97 \( 1 - 17.7iT - 97T^{2} \)
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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

−11.88194826539944770432666298754, −10.66297493969649816608852068641, −8.993045758232649597556043414053, −8.269184778599947585684831978241, −7.58307063872002838859823413274, −6.71315529618514205484033352683, −5.67101497756650538580301598858, −4.44446040658721308294122866010, −2.45936741652918426026068729313, −0.51740938965075097582979354074, 2.82767496822649744901696565273, 3.80151973940088265798531708972, 4.60853068282530423878700509070, 5.84288546204461390627691453966, 7.58846765967267509232173264744, 8.571241161373884556453383403485, 9.704617712353802040676177855729, 10.42520506684118839390343270737, 10.95286080965143623233923825342, 11.72275589461541830797216924002

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