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Properties

Label 2-290-145.102-c1-0-14
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: $\chi_{290} (247, \cdot )$
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

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

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