Properties

Label 2-230-115.109-c2-0-21
Degree $2$
Conductor $230$
Sign $0.681 + 0.732i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 + 0.587i)2-s + (1.02 − 3.48i)3-s + (1.30 + 1.51i)4-s + (4.98 − 0.329i)5-s + (3.36 − 3.87i)6-s + (1.08 − 7.54i)7-s + (0.796 + 2.71i)8-s + (−3.51 − 2.25i)9-s + (6.61 + 2.50i)10-s + (−6.50 + 2.97i)11-s + (6.60 − 3.01i)12-s + (−7.13 + 1.02i)13-s + (5.82 − 9.06i)14-s + (3.95 − 17.7i)15-s + (−0.569 + 3.95i)16-s + (−5.52 + 6.37i)17-s + ⋯
L(s)  = 1  + (0.643 + 0.293i)2-s + (0.340 − 1.16i)3-s + (0.327 + 0.377i)4-s + (0.997 − 0.0659i)5-s + (0.560 − 0.646i)6-s + (0.154 − 1.07i)7-s + (0.0996 + 0.339i)8-s + (−0.390 − 0.250i)9-s + (0.661 + 0.250i)10-s + (−0.591 + 0.270i)11-s + (0.550 − 0.251i)12-s + (−0.548 + 0.0788i)13-s + (0.416 − 0.647i)14-s + (0.263 − 1.18i)15-s + (−0.0355 + 0.247i)16-s + (−0.324 + 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.681 + 0.732i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.681 + 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.58847 - 1.12707i\)
\(L(\frac12)\) \(\approx\) \(2.58847 - 1.12707i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.587i)T \)
5 \( 1 + (-4.98 + 0.329i)T \)
23 \( 1 + (-19.1 - 12.7i)T \)
good3 \( 1 + (-1.02 + 3.48i)T + (-7.57 - 4.86i)T^{2} \)
7 \( 1 + (-1.08 + 7.54i)T + (-47.0 - 13.8i)T^{2} \)
11 \( 1 + (6.50 - 2.97i)T + (79.2 - 91.4i)T^{2} \)
13 \( 1 + (7.13 - 1.02i)T + (162. - 47.6i)T^{2} \)
17 \( 1 + (5.52 - 6.37i)T + (-41.1 - 286. i)T^{2} \)
19 \( 1 + (-5.13 + 4.45i)T + (51.3 - 357. i)T^{2} \)
29 \( 1 + (0.932 - 1.07i)T + (-119. - 832. i)T^{2} \)
31 \( 1 + (6.94 - 2.03i)T + (808. - 519. i)T^{2} \)
37 \( 1 + (36.5 + 23.4i)T + (568. + 1.24e3i)T^{2} \)
41 \( 1 + (39.5 - 25.4i)T + (698. - 1.52e3i)T^{2} \)
43 \( 1 + (-29.5 - 8.68i)T + (1.55e3 + 9.99e2i)T^{2} \)
47 \( 1 - 9.78iT - 2.20e3T^{2} \)
53 \( 1 + (11.3 - 78.9i)T + (-2.69e3 - 791. i)T^{2} \)
59 \( 1 + (-6.75 - 46.9i)T + (-3.33e3 + 980. i)T^{2} \)
61 \( 1 + (-24.4 - 83.1i)T + (-3.13e3 + 2.01e3i)T^{2} \)
67 \( 1 + (-8.95 + 19.6i)T + (-2.93e3 - 3.39e3i)T^{2} \)
71 \( 1 + (15.0 - 33.0i)T + (-3.30e3 - 3.80e3i)T^{2} \)
73 \( 1 + (52.8 - 45.7i)T + (758. - 5.27e3i)T^{2} \)
79 \( 1 + (-27.7 + 3.99i)T + (5.98e3 - 1.75e3i)T^{2} \)
83 \( 1 + (94.6 + 60.8i)T + (2.86e3 + 6.26e3i)T^{2} \)
89 \( 1 + (-14.2 + 48.6i)T + (-6.66e3 - 4.28e3i)T^{2} \)
97 \( 1 + (53.0 - 34.0i)T + (3.90e3 - 8.55e3i)T^{2} \)
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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

−12.32619544129716238863571072308, −10.94929468844290876452531707438, −10.01562044409620377190499949650, −8.662008759932559598983275771843, −7.35461270045621557091583755995, −7.05028944008184051634657298522, −5.74176458614846988616484983133, −4.54445869528050592559101296378, −2.76257485907762492303211541824, −1.47319575663636682493391216980, 2.25340911166339072546815133547, 3.28742737580848261758197064593, 4.92652628625246276407260922945, 5.40522737284040643174432180362, 6.75875040476300272108855732706, 8.558522652990647817352050979711, 9.406741654333318464042720480420, 10.16799119641036927387494194847, 10.99380802893569098024542469589, 12.17974079621477229387991101456

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