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Properties

Label 2-430-215.49-c1-0-20
Degree $2$
Conductor $430$
Sign $-0.896 - 0.442i$
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  − i·2-s + (−0.409 − 0.236i)3-s − 4-s + (−1.79 + 1.32i)5-s + (−0.236 + 0.409i)6-s + (2.61 − 1.50i)7-s + i·8-s + (−1.38 − 2.40i)9-s + (1.32 + 1.79i)10-s − 5.53·11-s + (0.409 + 0.236i)12-s + (−0.974 + 0.562i)13-s + (−1.50 − 2.61i)14-s + (1.05 − 0.119i)15-s + 16-s + (−1.03 + 0.594i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.236 − 0.136i)3-s − 0.5·4-s + (−0.804 + 0.594i)5-s + (−0.0966 + 0.167i)6-s + (0.987 − 0.570i)7-s + 0.353i·8-s + (−0.462 − 0.801i)9-s + (0.420 + 0.568i)10-s − 1.66·11-s + (0.118 + 0.0683i)12-s + (−0.270 + 0.155i)13-s + (−0.403 − 0.698i)14-s + (0.271 − 0.0308i)15-s + 0.250·16-s + (−0.249 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.896 - 0.442i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.896 - 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0584292 + 0.250345i\)
\(L(\frac12)\) \(\approx\) \(0.0584292 + 0.250345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (1.79 - 1.32i)T \)
43 \( 1 + (-6.46 + 1.11i)T \)
good3 \( 1 + (0.409 + 0.236i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.61 + 1.50i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.53T + 11T^{2} \)
13 \( 1 + (0.974 - 0.562i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.03 - 0.594i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 - 2.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.18 + 4.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.27 - 7.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.43 + 0.825i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.16T + 41T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (-3.64 - 2.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + (-2.32 - 4.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.23 + 2.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.30 + 2.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.10 + 5.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.19 + 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.3 + 7.10i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.95 + 5.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.61iT - 97T^{2} \)
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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

−10.53918352170448576474459271300, −10.34540903610286177788022752161, −8.665933416438551850336905308332, −7.983136927318900889513827896440, −7.09416485938221859245236875137, −5.71935207174434579920307219943, −4.54775827379452469511511851949, −3.54836752950898954569077186317, −2.22373654786443584672833817210, −0.15841497138236308524013878514, 2.36586548367930950015810982026, 4.22109968447382763745447741811, 5.22790643907243562644181698247, 5.57558734646146279002802371487, 7.41243488057671190163698985621, 8.022431816665637171636458189017, 8.523915020186812262123554233281, 9.780691441121676351581317688712, 10.98508333670466575724160495187, 11.50781052509555383450682139313

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