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Properties

Label 2-290-145.108-c1-0-9
Degree $2$
Conductor $290$
Sign $0.715 - 0.698i$
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  + (−0.433 − 0.900i)2-s + (1.96 + 2.46i)3-s + (−0.623 + 0.781i)4-s + (1.97 + 1.04i)5-s + (1.36 − 2.84i)6-s + (1.37 + 0.154i)7-s + (0.974 + 0.222i)8-s + (−1.54 + 6.78i)9-s + (0.0807 − 2.23i)10-s + (−2.67 − 1.68i)11-s − 3.15·12-s + (−5.15 − 3.24i)13-s + (−0.455 − 1.30i)14-s + (1.32 + 6.93i)15-s + (−0.222 − 0.974i)16-s − 3.56i·17-s + ⋯
L(s)  = 1  + (−0.306 − 0.637i)2-s + (1.13 + 1.42i)3-s + (−0.311 + 0.390i)4-s + (0.884 + 0.466i)5-s + (0.559 − 1.16i)6-s + (0.518 + 0.0584i)7-s + (0.344 + 0.0786i)8-s + (−0.516 + 2.26i)9-s + (0.0255 − 0.706i)10-s + (−0.807 − 0.507i)11-s − 0.911·12-s + (−1.43 − 0.898i)13-s + (−0.121 − 0.348i)14-s + (0.341 + 1.78i)15-s + (−0.0556 − 0.243i)16-s − 0.865i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ 0.715 - 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55027 + 0.631388i\)
\(L(\frac12)\) \(\approx\) \(1.55027 + 0.631388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (-1.97 - 1.04i)T \)
29 \( 1 + (-3.50 + 4.08i)T \)
good3 \( 1 + (-1.96 - 2.46i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 + (-1.37 - 0.154i)T + (6.82 + 1.55i)T^{2} \)
11 \( 1 + (2.67 + 1.68i)T + (4.77 + 9.91i)T^{2} \)
13 \( 1 + (5.15 + 3.24i)T + (5.64 + 11.7i)T^{2} \)
17 \( 1 + 3.56iT - 17T^{2} \)
19 \( 1 + (-2.84 + 0.320i)T + (18.5 - 4.22i)T^{2} \)
23 \( 1 + (-0.180 - 0.517i)T + (-17.9 + 14.3i)T^{2} \)
31 \( 1 + (-1.54 + 4.42i)T + (-24.2 - 19.3i)T^{2} \)
37 \( 1 + (1.26 - 5.55i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.56 - 4.56i)T + 41iT^{2} \)
43 \( 1 + (-9.78 - 4.71i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.66 + 7.31i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.0956 - 0.0334i)T + (41.4 + 33.0i)T^{2} \)
59 \( 1 + 10.8iT - 59T^{2} \)
61 \( 1 + (5.81 + 0.655i)T + (59.4 + 13.5i)T^{2} \)
67 \( 1 + (6.43 - 4.04i)T + (29.0 - 60.3i)T^{2} \)
71 \( 1 + (-1.35 + 0.310i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.41 - 13.3i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (-0.248 + 0.156i)T + (34.2 - 71.1i)T^{2} \)
83 \( 1 + (8.84 - 0.996i)T + (80.9 - 18.4i)T^{2} \)
89 \( 1 + (-2.25 - 0.790i)T + (69.5 + 55.4i)T^{2} \)
97 \( 1 + (11.0 - 13.8i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.51701320472619703925140368247, −10.64062433333524042317791096252, −9.824401595644717717173115821732, −9.571079465098082702440235730525, −8.308863518762067159483339892558, −7.58289794505448800091726471532, −5.42068538331374530194568235951, −4.63761788092735334308223531246, −2.98301344512181918944993980529, −2.55292110523493700953224926566, 1.52767802744526518221256756453, 2.53342974425055625537136672778, 4.71178630636080044916605192602, 5.99067165570484032830448430098, 7.16346201174310958502000299727, 7.65768094514964268185930288656, 8.736304854389722360110487869271, 9.333454285919642350714646535106, 10.43589418964899941317439491863, 12.27099407744423769917393217636

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