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Properties

Label 2-290-145.118-c1-0-1
Degree $2$
Conductor $290$
Sign $0.151 - 0.988i$
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  + (0.781 − 0.623i)2-s + (0.677 + 2.96i)3-s + (0.222 − 0.974i)4-s + (−1.61 + 1.54i)5-s + (2.37 + 1.89i)6-s + (0.822 + 0.516i)7-s + (−0.433 − 0.900i)8-s + (−5.64 + 2.71i)9-s + (−0.294 + 2.21i)10-s + (1.75 − 0.613i)11-s + 3.04·12-s + (−0.582 + 0.203i)13-s + (0.965 − 0.108i)14-s + (−5.68 − 3.73i)15-s + (−0.900 − 0.433i)16-s + 1.38i·17-s + ⋯
L(s)  = 1  + (0.552 − 0.440i)2-s + (0.390 + 1.71i)3-s + (0.111 − 0.487i)4-s + (−0.721 + 0.692i)5-s + (0.971 + 0.774i)6-s + (0.310 + 0.195i)7-s + (−0.153 − 0.318i)8-s + (−1.88 + 0.905i)9-s + (−0.0932 + 0.700i)10-s + (0.528 − 0.184i)11-s + 0.878·12-s + (−0.161 + 0.0565i)13-s + (0.258 − 0.0290i)14-s + (−1.46 − 0.964i)15-s + (−0.225 − 0.108i)16-s + 0.335i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ 0.151 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30586 + 1.12097i\)
\(L(\frac12)\) \(\approx\) \(1.30586 + 1.12097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (1.61 - 1.54i)T \)
29 \( 1 + (-4.05 - 3.54i)T \)
good3 \( 1 + (-0.677 - 2.96i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 + (-0.822 - 0.516i)T + (3.03 + 6.30i)T^{2} \)
11 \( 1 + (-1.75 + 0.613i)T + (8.60 - 6.85i)T^{2} \)
13 \( 1 + (0.582 - 0.203i)T + (10.1 - 8.10i)T^{2} \)
17 \( 1 - 1.38iT - 17T^{2} \)
19 \( 1 + (-0.0458 + 0.0287i)T + (8.24 - 17.1i)T^{2} \)
23 \( 1 + (-8.54 + 0.962i)T + (22.4 - 5.11i)T^{2} \)
31 \( 1 + (8.96 + 1.01i)T + (30.2 + 6.89i)T^{2} \)
37 \( 1 + (-8.02 + 3.86i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-0.177 - 0.177i)T + 41iT^{2} \)
43 \( 1 + (-3.13 + 3.93i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-4.95 - 2.38i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-0.703 + 6.24i)T + (-51.6 - 11.7i)T^{2} \)
59 \( 1 + 1.11iT - 59T^{2} \)
61 \( 1 + (-11.9 - 7.49i)T + (26.4 + 54.9i)T^{2} \)
67 \( 1 + (8.32 + 2.91i)T + (52.3 + 41.7i)T^{2} \)
71 \( 1 + (-1.51 + 3.14i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (9.85 + 7.85i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (14.1 + 4.93i)T + (61.7 + 49.2i)T^{2} \)
83 \( 1 + (13.9 - 8.77i)T + (36.0 - 74.7i)T^{2} \)
89 \( 1 + (1.71 - 15.2i)T + (-86.7 - 19.8i)T^{2} \)
97 \( 1 + (0.152 - 0.668i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.61526075823210379535171160706, −11.01569688087611829266980900229, −10.40446289846605595447090092810, −9.310876650836762365309662205212, −8.554204003552207153250534822978, −7.10070725988148749400544578352, −5.59487150473104519540817807293, −4.52280438316804356548551475123, −3.71014623111423485761768851451, −2.77540819013441701179185260224, 1.19107229925133597453551004360, 2.90598315605477906030659328082, 4.41756290365620772980307752348, 5.72093241051967826470378547368, 6.99299734953483796365321277017, 7.46713654426480634718292602989, 8.383082879817637832191260853588, 9.183716438667017545877243550540, 11.28289512435776731605278818943, 11.84065453738579916661360962175

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