Properties

Label 2-230-115.22-c1-0-5
Degree $2$
Conductor $230$
Sign $-0.684 + 0.728i$
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  + (−0.707 − 0.707i)2-s + (−1.96 + 1.96i)3-s + 1.00i·4-s + (−1.77 + 1.36i)5-s + 2.77·6-s + (−0.105 + 0.105i)7-s + (0.707 − 0.707i)8-s − 4.69i·9-s + (2.21 + 0.292i)10-s − 6.18i·11-s + (−1.96 − 1.96i)12-s + (−2.81 + 2.81i)13-s + 0.149·14-s + (0.812 − 6.15i)15-s − 1.00·16-s + (3.81 − 3.81i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.13 + 1.13i)3-s + 0.500i·4-s + (−0.793 + 0.608i)5-s + 1.13·6-s + (−0.0398 + 0.0398i)7-s + (0.250 − 0.250i)8-s − 1.56i·9-s + (0.701 + 0.0926i)10-s − 1.86i·11-s + (−0.566 − 0.566i)12-s + (−0.781 + 0.781i)13-s + 0.0398·14-s + (0.209 − 1.58i)15-s − 0.250·16-s + (0.924 − 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.684 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0302526 - 0.0699438i\)
\(L(\frac12)\) \(\approx\) \(0.0302526 - 0.0699438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.77 - 1.36i)T \)
23 \( 1 + (4.42 - 1.84i)T \)
good3 \( 1 + (1.96 - 1.96i)T - 3iT^{2} \)
7 \( 1 + (0.105 - 0.105i)T - 7iT^{2} \)
11 \( 1 + 6.18iT - 11T^{2} \)
13 \( 1 + (2.81 - 2.81i)T - 13iT^{2} \)
17 \( 1 + (-3.81 + 3.81i)T - 17iT^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
29 \( 1 - 0.693iT - 29T^{2} \)
31 \( 1 + 1.47T + 31T^{2} \)
37 \( 1 + (3.09 - 3.09i)T - 37iT^{2} \)
41 \( 1 - 3.87T + 41T^{2} \)
43 \( 1 + (7.58 + 7.58i)T + 43iT^{2} \)
47 \( 1 + (3.50 + 3.50i)T + 47iT^{2} \)
53 \( 1 + (1.81 + 1.81i)T + 53iT^{2} \)
59 \( 1 + 2.80iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + (6.69 - 6.69i)T - 67iT^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + (-7.61 + 7.61i)T - 73iT^{2} \)
79 \( 1 - 8.97T + 79T^{2} \)
83 \( 1 + (0.0424 + 0.0424i)T + 83iT^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + (11.1 - 11.1i)T - 97iT^{2} \)
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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

−11.70756757640553122186116338502, −10.87601725207208411489371489579, −10.22924381098657776161285494765, −9.206681298207302244751399903165, −8.044611295529285786972669766562, −6.70070068979807752880440513567, −5.51010460819668801045647775214, −4.19108150440849019984355766598, −3.17998828058368920199366373502, −0.083682737305110812155646850750, 1.69615811070278699054642596532, 4.50252590534982063385246185395, 5.53904282092374963019070348989, 6.69725792574971456262906021227, 7.57356486447490183379496999871, 8.133995085267940629010171786743, 9.745190753495402033631384635817, 10.64225471181715765275391083010, 11.89247690859456859636017779088, 12.51284095963065231604161395507

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