Properties

Label 2-430-43.4-c1-0-3
Degree $2$
Conductor $430$
Sign $-0.487 - 0.872i$
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  + (−0.623 + 0.781i)2-s + (1.81 + 2.27i)3-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s − 2.90·6-s + 1.09·7-s + (0.900 + 0.433i)8-s + (−1.21 + 5.32i)9-s + (−0.900 + 0.433i)10-s + (0.0810 − 0.354i)11-s + (1.81 − 2.27i)12-s + (−0.370 − 0.178i)13-s + (−0.684 + 0.858i)14-s + (0.647 + 2.83i)15-s + (−0.900 + 0.433i)16-s + (−1.53 + 0.741i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (1.04 + 1.31i)3-s + (−0.111 − 0.487i)4-s + (0.402 + 0.194i)5-s − 1.18·6-s + 0.415·7-s + (0.318 + 0.153i)8-s + (−0.404 + 1.77i)9-s + (−0.284 + 0.137i)10-s + (0.0244 − 0.107i)11-s + (0.523 − 0.656i)12-s + (−0.102 − 0.0494i)13-s + (−0.183 + 0.229i)14-s + (0.167 + 0.732i)15-s + (−0.225 + 0.108i)16-s + (−0.373 + 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.487 - 0.872i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839644 + 1.43091i\)
\(L(\frac12)\) \(\approx\) \(0.839644 + 1.43091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (4.73 + 4.53i)T \)
good3 \( 1 + (-1.81 - 2.27i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 + (-0.0810 + 0.354i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.370 + 0.178i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (1.53 - 0.741i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + (-1.31 - 5.77i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-1.80 + 7.88i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.405 - 0.509i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (0.0885 - 0.111i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 + (-5.84 + 7.32i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (0.499 + 2.18i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-7.24 + 3.49i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (7.01 - 3.37i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.62 - 4.54i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (0.571 + 2.50i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (0.00547 + 0.0240i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-14.1 - 6.80i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (6.22 + 7.80i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-4.79 - 6.01i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (0.383 - 1.67i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.86111547535415116830230742302, −10.35304000707933134620714032525, −9.587819201919994559496496331056, −8.681422611735467615726034542410, −8.207593142984679478322485288659, −6.96833690348993059601197746923, −5.64610525046215526028432413624, −4.65342916273438710152289072144, −3.56870278416097916655578323636, −2.17101952626870139434008740042, 1.24089871987676799175064738644, 2.27305350556814511123072847290, 3.31381665912977324684326291819, 4.97082962807333543960230564905, 6.54751285711706684245526077903, 7.41367229920788575535668005347, 8.128381566017085805407451482873, 9.093805179693019286312089742739, 9.570165468077925837504590407230, 11.04017587114008484722699317175

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