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Properties

Label 2-430-43.4-c1-0-10
Degree $2$
Conductor $430$
Sign $-0.427 + 0.903i$
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 + (−0.0568 − 0.0712i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s − 0.0911·6-s + 0.481·7-s + (−0.900 − 0.433i)8-s + (0.665 − 2.91i)9-s + (−0.900 + 0.433i)10-s + (0.483 − 2.11i)11-s + (−0.0568 + 0.0712i)12-s + (−1.12 − 0.541i)13-s + (0.300 − 0.376i)14-s + (0.0202 + 0.0888i)15-s + (−0.900 + 0.433i)16-s + (−0.238 + 0.114i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (−0.0327 − 0.0411i)3-s + (−0.111 − 0.487i)4-s + (−0.402 − 0.194i)5-s − 0.0371·6-s + 0.182·7-s + (−0.318 − 0.153i)8-s + (0.221 − 0.972i)9-s + (−0.284 + 0.137i)10-s + (0.145 − 0.639i)11-s + (−0.0163 + 0.0205i)12-s + (−0.311 − 0.150i)13-s + (0.0803 − 0.100i)14-s + (0.00523 + 0.0229i)15-s + (−0.225 + 0.108i)16-s + (−0.0579 + 0.0278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.427 + 0.903i$
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: \((2,\ 430,\ (\ :1/2),\ -0.427 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794326 - 1.25501i\)
\(L(\frac12)\) \(\approx\) \(0.794326 - 1.25501i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.947 - 6.48i)T \)
good3 \( 1 + (0.0568 + 0.0712i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 - 0.481T + 7T^{2} \)
11 \( 1 + (-0.483 + 2.11i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.12 + 0.541i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (0.238 - 0.114i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + (0.805 + 3.52i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-0.757 + 3.32i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.06 + 2.58i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.83 - 3.55i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 - 6.79T + 37T^{2} \)
41 \( 1 + (0.298 - 0.373i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-2.07 - 9.07i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-11.3 + 5.47i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-6.47 + 3.11i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.28 - 4.11i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-1.47 - 6.47i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.16 - 5.10i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.919 + 0.442i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 7.50T + 79T^{2} \)
83 \( 1 + (-3.90 - 4.89i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (9.36 + 11.7i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (0.935 - 4.09i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.15303495673881159614803308709, −10.05673782635609494307851357850, −9.149250611586948548847506820267, −8.287617934481374917955993145816, −6.98798716059642178956740131979, −6.06797209096595141506080075623, −4.83625710133446940329462154643, −3.88348489947931071519168908366, −2.72963861506687385139371371444, −0.864243820299388620046807698148, 2.15355791933520481127406370851, 3.74507988443926203143474444405, 4.71867853480292879346733336701, 5.64928639471813693217639701666, 6.94840952084533869267399933317, 7.60499943462591741252135803801, 8.451344993422094279923532080182, 9.662696325443668220577507119734, 10.60046621621144200853722474041, 11.56377376795611601168429901439

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