Properties

Label 2-230-5.4-c5-0-44
Degree $2$
Conductor $230$
Sign $0.595 + 0.803i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 5.15i·3-s − 16·4-s + (33.2 + 44.9i)5-s + 20.6·6-s − 47.7i·7-s − 64i·8-s + 216.·9-s + (−179. + 133. i)10-s − 454.·11-s + 82.4i·12-s − 981. i·13-s + 190.·14-s + (231. − 171. i)15-s + 256·16-s − 62.6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.330i·3-s − 0.5·4-s + (0.595 + 0.803i)5-s + 0.233·6-s − 0.368i·7-s − 0.353i·8-s + 0.890·9-s + (−0.568 + 0.421i)10-s − 1.13·11-s + 0.165i·12-s − 1.61i·13-s + 0.260·14-s + (0.265 − 0.196i)15-s + 0.250·16-s − 0.0525i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.440170429\)
\(L(\frac12)\) \(\approx\) \(1.440170429\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (-33.2 - 44.9i)T \)
23 \( 1 - 529iT \)
good3 \( 1 + 5.15iT - 243T^{2} \)
7 \( 1 + 47.7iT - 1.68e4T^{2} \)
11 \( 1 + 454.T + 1.61e5T^{2} \)
13 \( 1 + 981. iT - 3.71e5T^{2} \)
17 \( 1 + 62.6iT - 1.41e6T^{2} \)
19 \( 1 + 779.T + 2.47e6T^{2} \)
29 \( 1 + 6.54e3T + 2.05e7T^{2} \)
31 \( 1 + 393.T + 2.86e7T^{2} \)
37 \( 1 + 1.33e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.81e4T + 1.15e8T^{2} \)
43 \( 1 + 1.09e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.25e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.56e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.68e4T + 7.14e8T^{2} \)
61 \( 1 + 2.56e4T + 8.44e8T^{2} \)
67 \( 1 + 4.60e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.08e4T + 1.80e9T^{2} \)
73 \( 1 + 4.70e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.66e4T + 3.07e9T^{2} \)
83 \( 1 - 4.86e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.75e4T + 5.58e9T^{2} \)
97 \( 1 + 1.69e5iT - 8.58e9T^{2} \)
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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

−10.73036319483555942876532912476, −10.38946152055681679112201874333, −9.245351775346299838429433476202, −7.66229244267267529939249029620, −7.43044892217931288473172314251, −6.09539949585733589852774295248, −5.25784395960841547890823778039, −3.66435259421767453549641579731, −2.21422020793338666688500523260, −0.42222836426975115063606590439, 1.37870529744855993710402887370, 2.41758454572335557454519487473, 4.18176784555945166628309677506, 4.91280253026126007827729765108, 6.16562927844772476944574707767, 7.69496629665709616740155637901, 8.968268078941804785042979093776, 9.530416215052357967134254783679, 10.44954896330628410473010666194, 11.44394068163507639075212595943

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