Properties

Label 2-230-23.13-c1-0-5
Degree $2$
Conductor $230$
Sign $0.482 - 0.875i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + (0.654 + 0.755i)2-s + (1.45 + 0.935i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (0.246 + 1.71i)6-s + (1.53 + 0.451i)7-s + (−0.841 + 0.540i)8-s + (−0.00358 − 0.00785i)9-s + (0.959 − 0.281i)10-s + (0.784 − 0.904i)11-s + (−1.13 + 1.30i)12-s + (−4.64 + 1.36i)13-s + (0.665 + 1.45i)14-s + (1.45 − 0.935i)15-s + (−0.959 − 0.281i)16-s + (0.194 + 1.35i)17-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (0.840 + 0.539i)3-s + (−0.0711 + 0.494i)4-s + (0.185 − 0.406i)5-s + (0.100 + 0.698i)6-s + (0.581 + 0.170i)7-s + (−0.297 + 0.191i)8-s + (−0.00119 − 0.00261i)9-s + (0.303 − 0.0890i)10-s + (0.236 − 0.272i)11-s + (−0.326 + 0.377i)12-s + (−1.28 + 0.377i)13-s + (0.177 + 0.389i)14-s + (0.375 − 0.241i)15-s + (−0.239 − 0.0704i)16-s + (0.0471 + 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.482 - 0.875i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.482 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69305 + 1.00012i\)
\(L(\frac12)\) \(\approx\) \(1.69305 + 1.00012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (4.66 + 1.09i)T \)
good3 \( 1 + (-1.45 - 0.935i)T + (1.24 + 2.72i)T^{2} \)
7 \( 1 + (-1.53 - 0.451i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-0.784 + 0.904i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (4.64 - 1.36i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.194 - 1.35i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.206 + 1.43i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.197 + 1.37i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.564 - 0.363i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-3.99 - 8.73i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-3.41 + 7.46i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (3.73 + 2.40i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + (-5.54 - 1.62i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (9.36 - 2.75i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (0.954 - 0.613i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (4.19 + 4.84i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-0.918 - 1.06i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.286 - 1.99i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (8.87 - 2.60i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-2.55 - 5.59i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.855 - 0.549i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (2.09 - 4.59i)T + (-63.5 - 73.3i)T^{2} \)
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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

−12.37244956461195268954397222004, −11.71056913930367841010530248056, −10.19989994440963010498572382374, −9.225644639119345631935161703320, −8.481583613699952672553395781934, −7.50902818804054684387355774492, −6.13632764328075802131209067580, −4.89642339516829722314428655217, −3.95350361181997872996783644726, −2.45675661595400020589026566218, 1.91119683904454900973190761912, 2.93085586917576744845777716827, 4.44555522422563407135363535038, 5.71783786184374502482298958953, 7.24289826222070239525915056923, 7.908179115833507274520515546386, 9.241865010237631565048093177335, 10.16770527883679713276048180423, 11.18248132116121374396632549956, 12.18739715533634541748961047949

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