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Properties

Label 2-430-215.42-c1-0-19
Degree $2$
Conductor $430$
Sign $-0.887 + 0.459i$
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 + (−1.53 − 1.53i)3-s − 1.00i·4-s + (1.06 − 1.96i)5-s − 2.16·6-s + (1.64 − 1.64i)7-s + (−0.707 − 0.707i)8-s + 1.69i·9-s + (−0.633 − 2.14i)10-s + 3.84·11-s + (−1.53 + 1.53i)12-s + (−2.55 + 2.55i)13-s − 2.32i·14-s + (−4.64 + 1.37i)15-s − 1.00·16-s + (−1.69 − 1.69i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.884 − 0.884i)3-s − 0.500i·4-s + (0.477 − 0.878i)5-s − 0.884·6-s + (0.622 − 0.622i)7-s + (−0.250 − 0.250i)8-s + 0.566i·9-s + (−0.200 − 0.678i)10-s + 1.15·11-s + (−0.442 + 0.442i)12-s + (−0.708 + 0.708i)13-s − 0.622i·14-s + (−1.20 + 0.354i)15-s − 0.250·16-s + (−0.410 − 0.410i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.887 + 0.459i$
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: \((2,\ 430,\ (\ :1/2),\ -0.887 + 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352551 - 1.44699i\)
\(L(\frac12)\) \(\approx\) \(0.352551 - 1.44699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 1.96i)T \)
43 \( 1 + (2.32 - 6.13i)T \)
good3 \( 1 + (1.53 + 1.53i)T + 3iT^{2} \)
7 \( 1 + (-1.64 + 1.64i)T - 7iT^{2} \)
11 \( 1 - 3.84T + 11T^{2} \)
13 \( 1 + (2.55 - 2.55i)T - 13iT^{2} \)
17 \( 1 + (1.69 + 1.69i)T + 17iT^{2} \)
19 \( 1 - 1.66T + 19T^{2} \)
23 \( 1 + (4.75 - 4.75i)T - 23iT^{2} \)
29 \( 1 + 3.07T + 29T^{2} \)
31 \( 1 - 9.51T + 31T^{2} \)
37 \( 1 + (-3.52 + 3.52i)T - 37iT^{2} \)
41 \( 1 - 1.30T + 41T^{2} \)
47 \( 1 + (-2.14 - 2.14i)T + 47iT^{2} \)
53 \( 1 + (-6.79 + 6.79i)T - 53iT^{2} \)
59 \( 1 + 5.14iT - 59T^{2} \)
61 \( 1 + 7.55iT - 61T^{2} \)
67 \( 1 + (-2.61 - 2.61i)T + 67iT^{2} \)
71 \( 1 - 7.21iT - 71T^{2} \)
73 \( 1 + (6.06 + 6.06i)T + 73iT^{2} \)
79 \( 1 - 15.6iT - 79T^{2} \)
83 \( 1 + (-12.4 + 12.4i)T - 83iT^{2} \)
89 \( 1 - 3.54T + 89T^{2} \)
97 \( 1 + (-7.33 - 7.33i)T + 97iT^{2} \)
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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

−11.31234732005490867412352055779, −9.886899042248775492514974241963, −9.232690318707695970662970373172, −7.83644250811411606028503969863, −6.78886257134367495557277180746, −5.97662144906562701248740810370, −4.93159903644481585482259147080, −4.06849465736516097021357465170, −1.93349076248370413739755022581, −0.971129169246508489842425092955, 2.43129557200803958694497065384, 3.95572222428399540186186301574, 4.92225334349204968328816678604, 5.85260546985746985968330342753, 6.47628192041045490602280390462, 7.70880339814815504867577761072, 8.871468602425693923864376103625, 10.00868934338921721411515237026, 10.58910411175548306216584938305, 11.73999691794226581658405076364

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