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Properties

Label 2-430-43.11-c1-0-1
Degree $2$
Conductor $430$
Sign $0.0762 - 0.997i$
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 + (−0.583 + 0.731i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + 0.935·6-s + 2.01·7-s + (0.900 − 0.433i)8-s + (0.472 + 2.07i)9-s + (0.900 + 0.433i)10-s + (−0.717 − 3.14i)11-s + (−0.583 − 0.731i)12-s + (−4.49 + 2.16i)13-s + (−1.25 − 1.57i)14-s + (0.208 − 0.912i)15-s + (−0.900 − 0.433i)16-s + (−0.0803 − 0.0387i)17-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (−0.336 + 0.422i)3-s + (−0.111 + 0.487i)4-s + (−0.402 + 0.194i)5-s + 0.381·6-s + 0.760·7-s + (0.318 − 0.153i)8-s + (0.157 + 0.690i)9-s + (0.284 + 0.137i)10-s + (−0.216 − 0.948i)11-s + (−0.168 − 0.211i)12-s + (−1.24 + 0.600i)13-s + (−0.335 − 0.420i)14-s + (0.0537 − 0.235i)15-s + (−0.225 − 0.108i)16-s + (−0.0194 − 0.00939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0762 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0762 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.0762 - 0.997i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.0762 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516734 + 0.478704i\)
\(L(\frac12)\) \(\approx\) \(0.516734 + 0.478704i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 6.10i)T \)
good3 \( 1 + (0.583 - 0.731i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 - 2.01T + 7T^{2} \)
11 \( 1 + (0.717 + 3.14i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (4.49 - 2.16i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (0.0803 + 0.0387i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + (1.60 - 7.05i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-0.858 - 3.75i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-3.80 - 4.77i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-4.73 - 5.93i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + 9.33T + 37T^{2} \)
41 \( 1 + (-3.08 - 3.86i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-1.86 + 8.19i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (5.38 + 2.59i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-3.61 - 1.74i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-9.46 + 11.8i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (2.34 - 10.2i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (2.93 - 12.8i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-10.4 + 5.05i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (-4.96 + 6.22i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-7.38 + 9.26i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (2.64 + 11.6i)T + (-87.3 + 42.0i)T^{2} \)
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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

−11.34158120757014417738993861806, −10.45236249941978148450666483476, −9.919895934958546869537743900825, −8.544236167146718792888270363782, −7.976879623109425577036850367310, −6.92383937086040615483611567118, −5.37330058984989121141701379055, −4.54373599326269939759084250294, −3.28234835627670829094569866708, −1.77379281866984468106951226016, 0.54123731097311182842195603108, 2.36561304657769457587977686461, 4.43313757715276036393680706487, 5.13970271207526022546571326950, 6.52071984360975586264003386687, 7.26805359133260849075895493189, 8.006598080150008432915101367245, 9.052561547674614118233155143173, 9.941587787284194497635796696913, 10.90549439774239027291121247319

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