↑

Properties

Label 2-430-215.42-c1-0-13
Degree $2$
Conductor $430$
Sign $-0.221 + 0.975i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.88 − 1.88i)3-s − 1.00i·4-s + (1.60 − 1.56i)5-s + 2.66·6-s + (3.34 − 3.34i)7-s + (0.707 + 0.707i)8-s + 4.10i·9-s + (−0.0281 + 2.23i)10-s − 0.253·11-s + (−1.88 + 1.88i)12-s + (0.923 − 0.923i)13-s + 4.72i·14-s + (−5.96 − 0.0749i)15-s − 1.00·16-s + (3.74 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−1.08 − 1.08i)3-s − 0.500i·4-s + (0.715 − 0.698i)5-s + 1.08·6-s + (1.26 − 1.26i)7-s + (0.250 + 0.250i)8-s + 1.36i·9-s + (−0.00888 + 0.707i)10-s − 0.0764·11-s + (−0.544 + 0.544i)12-s + (0.256 − 0.256i)13-s + 1.26i·14-s + (−1.53 − 0.0193i)15-s − 0.250·16-s + (0.909 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.581491 - 0.728432i\)
\(L(\frac12)\) \(\approx\) \(0.581491 - 0.728432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 + 1.56i)T \)
43 \( 1 + (4.73 + 4.53i)T \)
good3 \( 1 + (1.88 + 1.88i)T + 3iT^{2} \)
7 \( 1 + (-3.34 + 3.34i)T - 7iT^{2} \)
11 \( 1 + 0.253T + 11T^{2} \)
13 \( 1 + (-0.923 + 0.923i)T - 13iT^{2} \)
17 \( 1 + (-3.74 - 3.74i)T + 17iT^{2} \)
19 \( 1 + 4.13T + 19T^{2} \)
23 \( 1 + (-5.72 + 5.72i)T - 23iT^{2} \)
29 \( 1 - 1.33T + 29T^{2} \)
31 \( 1 + 2.79T + 31T^{2} \)
37 \( 1 + (2.31 - 2.31i)T - 37iT^{2} \)
41 \( 1 + 1.30T + 41T^{2} \)
47 \( 1 + (0.243 + 0.243i)T + 47iT^{2} \)
53 \( 1 + (8.47 - 8.47i)T - 53iT^{2} \)
59 \( 1 - 6.95iT - 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 + (-5.05 - 5.05i)T + 67iT^{2} \)
71 \( 1 + 1.69iT - 71T^{2} \)
73 \( 1 + (5.43 + 5.43i)T + 73iT^{2} \)
79 \( 1 - 7.00iT - 79T^{2} \)
83 \( 1 + (-7.42 + 7.42i)T - 83iT^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (5.67 + 5.67i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.56591671728802641255873321669, −10.49999119580132300273690457634, −8.756788992959987384081091445355, −8.022435459332281839359495329261, −7.14742380676430771313384656508, −6.28492914995565358474120212990, −5.37935197866288049244581660215, −4.50066596994256734329605687102, −1.68279753728553369213966550937, −0.886815582893764926388627446502, 1.86096695193003280920468306601, 3.29771847061711336410408536939, 4.93475259717065609244339370497, 5.41543292688749051712505850665, 6.54963379839027658868518618321, 7.948690513498656298579386358710, 9.148039440263601521742679641458, 9.677611037698932819698287597830, 10.70994406983093828659781841893, 11.27604715140642289683385883165

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