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Properties

Label 2-430-215.128-c1-0-12
Degree $2$
Conductor $430$
Sign $-0.170 + 0.985i$
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  + (−0.707 − 0.707i)2-s + (−1.41 + 1.41i)3-s + 1.00i·4-s + (−0.707 + 2.12i)5-s + 2.00·6-s + (−2.82 − 2.82i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (2 − 0.999i)10-s − 4·11-s + (−1.41 − 1.41i)12-s + (2 + 2i)13-s + 4.00i·14-s + (−1.99 − 4i)15-s − 1.00·16-s + (5 − 5i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.816 + 0.816i)3-s + 0.500i·4-s + (−0.316 + 0.948i)5-s + 0.816·6-s + (−1.06 − 1.06i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.632 − 0.316i)10-s − 1.20·11-s + (−0.408 − 0.408i)12-s + (0.554 + 0.554i)13-s + 1.06i·14-s + (−0.516 − 1.03i)15-s − 0.250·16-s + (1.21 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.170 + 0.985i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.170 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183766 - 0.218372i\)
\(L(\frac12)\) \(\approx\) \(0.183766 - 0.218372i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 2.12i)T \)
43 \( 1 + (-0.535 + 6.53i)T \)
good3 \( 1 + (1.41 - 1.41i)T - 3iT^{2} \)
7 \( 1 + (2.82 + 2.82i)T + 7iT^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (4.24 + 4.24i)T + 37iT^{2} \)
41 \( 1 + 12T + 41T^{2} \)
47 \( 1 + (5 - 5i)T - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 4.24iT - 61T^{2} \)
67 \( 1 + (2 - 2i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (-1.41 + 1.41i)T - 73iT^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + (-6 - 6i)T + 83iT^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-9 + 9i)T - 97iT^{2} \)
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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

−10.57028597129474452107205629824, −10.21848854346127369239996281188, −9.733448745853539517941585221933, −8.094629354658803420988745614473, −7.22267585827641568428322435734, −6.29776237429667155659789947250, −4.95197989284377843706270277186, −3.75415106349849940817914898747, −2.87449900030571154165815576195, −0.25656437065591258210781784916, 1.30918334681467185394194591335, 3.28756673497654838920593747001, 5.33432562851094761789494607335, 5.72050041536769906331092709983, 6.61535021826330573440236830483, 7.966949031932369193881891996047, 8.321981672343759566765583039392, 9.643831504772539117619646643162, 10.26212128952693825256722729791, 11.80492585904344841567055214105

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