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Properties

Label 2-290-145.102-c1-0-7
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: $\chi_{290} (247, \cdot )$
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

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

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