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Properties

Label 2-290-145.118-c1-0-3
Degree $2$
Conductor $290$
Sign $0.930 - 0.367i$
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 + (−0.156 − 0.684i)3-s + (0.222 − 0.974i)4-s + (1.63 + 1.52i)5-s + (0.548 + 0.437i)6-s + (−0.532 − 0.334i)7-s + (0.433 + 0.900i)8-s + (2.25 − 1.08i)9-s + (−2.22 − 0.179i)10-s + (−0.949 + 0.332i)11-s − 0.701·12-s + (0.847 − 0.296i)13-s + (0.624 − 0.0703i)14-s + (0.792 − 1.35i)15-s + (−0.900 − 0.433i)16-s + 4.77i·17-s + ⋯
L(s)  = 1  + (−0.552 + 0.440i)2-s + (−0.0901 − 0.394i)3-s + (0.111 − 0.487i)4-s + (0.729 + 0.684i)5-s + (0.223 + 0.178i)6-s + (−0.201 − 0.126i)7-s + (0.153 + 0.318i)8-s + (0.753 − 0.362i)9-s + (−0.704 − 0.0567i)10-s + (−0.286 + 0.100i)11-s − 0.202·12-s + (0.235 − 0.0822i)13-s + (0.166 − 0.0188i)14-s + (0.204 − 0.349i)15-s + (−0.225 − 0.108i)16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $0.930 - 0.367i$
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: \((2,\ 290,\ (\ :1/2),\ 0.930 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10957 + 0.211229i\)
\(L(\frac12)\) \(\approx\) \(1.10957 + 0.211229i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 1.52i)T \)
29 \( 1 + (-5.22 - 1.29i)T \)
good3 \( 1 + (0.156 + 0.684i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 + (0.532 + 0.334i)T + (3.03 + 6.30i)T^{2} \)
11 \( 1 + (0.949 - 0.332i)T + (8.60 - 6.85i)T^{2} \)
13 \( 1 + (-0.847 + 0.296i)T + (10.1 - 8.10i)T^{2} \)
17 \( 1 - 4.77iT - 17T^{2} \)
19 \( 1 + (-6.11 + 3.84i)T + (8.24 - 17.1i)T^{2} \)
23 \( 1 + (-4.37 + 0.492i)T + (22.4 - 5.11i)T^{2} \)
31 \( 1 + (-0.125 - 0.0141i)T + (30.2 + 6.89i)T^{2} \)
37 \( 1 + (2.22 - 1.06i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-2.24 - 2.24i)T + 41iT^{2} \)
43 \( 1 + (4.15 - 5.21i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (10.3 + 5.00i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-0.209 + 1.85i)T + (-51.6 - 11.7i)T^{2} \)
59 \( 1 - 7.31iT - 59T^{2} \)
61 \( 1 + (10.3 + 6.49i)T + (26.4 + 54.9i)T^{2} \)
67 \( 1 + (7.48 + 2.61i)T + (52.3 + 41.7i)T^{2} \)
71 \( 1 + (-3.13 + 6.50i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.638 - 0.509i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (14.0 + 4.93i)T + (61.7 + 49.2i)T^{2} \)
83 \( 1 + (-5.71 + 3.59i)T + (36.0 - 74.7i)T^{2} \)
89 \( 1 + (-0.296 + 2.63i)T + (-86.7 - 19.8i)T^{2} \)
97 \( 1 + (2.42 - 10.6i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.73140483228316708865420109940, −10.64201670150698622512101497074, −9.964161259327742249535120592949, −9.106850306617625739282335100883, −7.83744560379455625877207220166, −6.87403030505857033728977700022, −6.30043927564485833537513108850, −5.00529698931046183114241157442, −3.15189628043975558557816309345, −1.45678852561233246349156762761, 1.36980722951478687246253338756, 3.00891464239781797755685607000, 4.59669790939604525027333484764, 5.54828651628260327645273321569, 7.00720955872155372583720845516, 8.090032726078509182696532678483, 9.257366230650309905594475672270, 9.761416296984021447358452265821, 10.58325239544499575744368051595, 11.68384485117916897217247548859

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