Properties

Label 2-230-115.97-c1-0-2
Degree $2$
Conductor $230$
Sign $-0.731 - 0.682i$
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  + (0.479 + 0.877i)2-s + (0.125 + 1.74i)3-s + (−0.540 + 0.841i)4-s + (−0.402 + 2.19i)5-s + (−1.47 + 0.948i)6-s + (0.229 + 0.0857i)7-s + (−0.997 − 0.0713i)8-s + (−0.0744 + 0.0106i)9-s + (−2.12 + 0.700i)10-s + (−1.48 − 5.06i)11-s + (−1.53 − 0.840i)12-s + (−0.499 − 1.33i)13-s + (0.0349 + 0.242i)14-s + (−3.89 − 0.429i)15-s + (−0.415 − 0.909i)16-s + (−1.44 + 6.62i)17-s + ⋯
L(s)  = 1  + (0.338 + 0.620i)2-s + (0.0722 + 1.00i)3-s + (−0.270 + 0.420i)4-s + (−0.180 + 0.983i)5-s + (−0.602 + 0.387i)6-s + (0.0869 + 0.0324i)7-s + (−0.352 − 0.0252i)8-s + (−0.0248 + 0.00356i)9-s + (−0.671 + 0.221i)10-s + (−0.448 − 1.52i)11-s + (−0.444 − 0.242i)12-s + (−0.138 − 0.371i)13-s + (0.00933 + 0.0649i)14-s + (−1.00 − 0.110i)15-s + (−0.103 − 0.227i)16-s + (−0.349 + 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.731 - 0.682i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.731 - 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508138 + 1.28976i\)
\(L(\frac12)\) \(\approx\) \(0.508138 + 1.28976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.479 - 0.877i)T \)
5 \( 1 + (0.402 - 2.19i)T \)
23 \( 1 + (-4.20 + 2.30i)T \)
good3 \( 1 + (-0.125 - 1.74i)T + (-2.96 + 0.426i)T^{2} \)
7 \( 1 + (-0.229 - 0.0857i)T + (5.29 + 4.58i)T^{2} \)
11 \( 1 + (1.48 + 5.06i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.499 + 1.33i)T + (-9.82 + 8.51i)T^{2} \)
17 \( 1 + (1.44 - 6.62i)T + (-15.4 - 7.06i)T^{2} \)
19 \( 1 + (-6.43 - 4.13i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (3.62 + 5.64i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (-2.28 - 2.63i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-1.69 - 1.26i)T + (10.4 + 35.5i)T^{2} \)
41 \( 1 + (-0.230 + 1.60i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-9.13 + 0.653i)T + (42.5 - 6.11i)T^{2} \)
47 \( 1 + (1.51 + 1.51i)T + 47iT^{2} \)
53 \( 1 + (0.284 - 0.762i)T + (-40.0 - 34.7i)T^{2} \)
59 \( 1 + (3.27 + 1.49i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-6.49 + 5.63i)T + (8.68 - 60.3i)T^{2} \)
67 \( 1 + (-10.0 + 5.47i)T + (36.2 - 56.3i)T^{2} \)
71 \( 1 + (4.61 + 1.35i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (9.02 - 1.96i)T + (66.4 - 30.3i)T^{2} \)
79 \( 1 + (1.23 - 2.70i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-7.53 + 10.0i)T + (-23.3 - 79.6i)T^{2} \)
89 \( 1 + (7.51 - 8.67i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (0.636 + 0.850i)T + (-27.3 + 93.0i)T^{2} \)
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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

−12.72707766094316668600553194211, −11.35659314908515561346559476487, −10.64262708074663075260195817287, −9.788384918414219028958096622011, −8.499513720385276933883886681764, −7.64087505634329700849986519122, −6.29619812678375843298434311955, −5.40805059760548020407947843456, −3.93430141611201065104060061101, −3.15073398107713192640970875909, 1.17192964859322024182945749921, 2.56333265149216553531405653603, 4.52474597058885639661487965959, 5.23302734061777311275840886499, 7.08749904638092702134211431333, 7.56607879201118071676140143927, 9.179953007904345457316875921609, 9.697062110179706489502362447783, 11.32288625228068416094068553770, 11.98921757455488398379218736794

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