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Properties

Label 2-230-115.114-c2-0-15
Degree $2$
Conductor $230$
Sign $0.786 + 0.617i$
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  + 1.41i·2-s + 0.894i·3-s − 2.00·4-s + (−3.14 − 3.88i)5-s − 1.26·6-s − 4.24·7-s − 2.82i·8-s + 8.19·9-s + (5.49 − 4.44i)10-s − 9.15i·11-s − 1.78i·12-s − 6.01i·13-s − 6.00i·14-s + (3.47 − 2.81i)15-s + 4.00·16-s + 19.3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.298i·3-s − 0.500·4-s + (−0.628 − 0.777i)5-s − 0.210·6-s − 0.606·7-s − 0.353i·8-s + 0.911·9-s + (0.549 − 0.444i)10-s − 0.832i·11-s − 0.149i·12-s − 0.462i·13-s − 0.428i·14-s + (0.231 − 0.187i)15-s + 0.250·16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.786 + 0.617i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01680 - 0.351627i\)
\(L(\frac12)\) \(\approx\) \(1.01680 - 0.351627i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (3.14 + 3.88i)T \)
23 \( 1 + (5.12 + 22.4i)T \)
good3 \( 1 - 0.894iT - 9T^{2} \)
7 \( 1 + 4.24T + 49T^{2} \)
11 \( 1 + 9.15iT - 121T^{2} \)
13 \( 1 + 6.01iT - 169T^{2} \)
17 \( 1 - 19.3T + 289T^{2} \)
19 \( 1 + 31.0iT - 361T^{2} \)
29 \( 1 + 45.5T + 841T^{2} \)
31 \( 1 - 33.3T + 961T^{2} \)
37 \( 1 + 21.2T + 1.36e3T^{2} \)
41 \( 1 - 33.7T + 1.68e3T^{2} \)
43 \( 1 - 5.50T + 1.84e3T^{2} \)
47 \( 1 + 71.8iT - 2.20e3T^{2} \)
53 \( 1 + 3.73T + 2.80e3T^{2} \)
59 \( 1 + 4.90T + 3.48e3T^{2} \)
61 \( 1 - 42.5iT - 3.72e3T^{2} \)
67 \( 1 + 118.T + 4.48e3T^{2} \)
71 \( 1 - 92.6T + 5.04e3T^{2} \)
73 \( 1 + 43.2iT - 5.32e3T^{2} \)
79 \( 1 + 58.6iT - 6.24e3T^{2} \)
83 \( 1 + 18.2T + 6.88e3T^{2} \)
89 \( 1 - 164. iT - 7.92e3T^{2} \)
97 \( 1 + 91.9T + 9.40e3T^{2} \)
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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

−12.08108365622826543509557071382, −10.81222473644235750401460139368, −9.711257930064469939192510891950, −8.868648412512319567309125617892, −7.88269159707013443273633321562, −6.90462531993795187022434978599, −5.60224188999116564372326176831, −4.54859535704675350063919066767, −3.40573944686072169516121465117, −0.62074663065994631652362148290, 1.68649665527226901116464499504, 3.33404797015793350100095475971, 4.25086153240409782266633547238, 5.98558104257132967030959594843, 7.25708617800935097759031422602, 7.907417050086197035508700763125, 9.668762784659177119134144676276, 10.02328705138338306500656577598, 11.19211349088565211167369620582, 12.23703461052362519831257332713

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