Properties

Label 2-430-5.4-c1-0-17
Degree $2$
Conductor $430$
Sign $-0.0700 + 0.997i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + i·2-s − 1.63i·3-s − 4-s + (0.156 − 2.23i)5-s + 1.63·6-s + 0.479i·7-s i·8-s + 0.312·9-s + (2.23 + 0.156i)10-s − 5.18·11-s + 1.63i·12-s − 6.25i·13-s − 0.479·14-s + (−3.65 − 0.256i)15-s + 16-s + 5.43i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.946i·3-s − 0.5·4-s + (0.0700 − 0.997i)5-s + 0.669·6-s + 0.181i·7-s − 0.353i·8-s + 0.104·9-s + (0.705 + 0.0495i)10-s − 1.56·11-s + 0.473i·12-s − 1.73i·13-s − 0.128·14-s + (−0.944 − 0.0663i)15-s + 0.250·16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0700 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0700 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.0700 + 0.997i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.0700 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710970 - 0.762652i\)
\(L(\frac12)\) \(\approx\) \(0.710970 - 0.762652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.156 + 2.23i)T \)
43 \( 1 + iT \)
good3 \( 1 + 1.63iT - 3T^{2} \)
7 \( 1 - 0.479iT - 7T^{2} \)
11 \( 1 + 5.18T + 11T^{2} \)
13 \( 1 + 6.25iT - 13T^{2} \)
17 \( 1 - 5.43iT - 17T^{2} \)
19 \( 1 + 1.83T + 19T^{2} \)
23 \( 1 + 6.87iT - 23T^{2} \)
29 \( 1 + 0.614T + 29T^{2} \)
31 \( 1 - 1.88T + 31T^{2} \)
37 \( 1 + 8.03iT - 37T^{2} \)
41 \( 1 - 6.73T + 41T^{2} \)
47 \( 1 - 8.58iT - 47T^{2} \)
53 \( 1 - 0.203iT - 53T^{2} \)
59 \( 1 - 3.90T + 59T^{2} \)
61 \( 1 - 4.93T + 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 - 4.66T + 71T^{2} \)
73 \( 1 - 1.06iT - 73T^{2} \)
79 \( 1 + 6.06T + 79T^{2} \)
83 \( 1 - 8.30iT - 83T^{2} \)
89 \( 1 - 9.89T + 89T^{2} \)
97 \( 1 + 16.1iT - 97T^{2} \)
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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

−10.71672768550558324069942580109, −10.05276121085443301854716417242, −8.599532423014331724284573945688, −8.099453641059305893182391580147, −7.44355649581807254172905119594, −6.06647304271436120511539336713, −5.46307632600722427455327270429, −4.28599327618616687746444056110, −2.42348497118358601509258028088, −0.65800763062668542490991861504, 2.21667873083881166932848543832, 3.35208399072101275330057918014, 4.40581058710709211231702401658, 5.33178545369987809927208689018, 6.82849596895769837511995669073, 7.70805245597936634815071790725, 9.146317557001573349919873555562, 9.816525980200328386798294427862, 10.46269927078645448974549756077, 11.24654394860670495897682788892

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