Properties

Label 2-430-43.36-c1-0-2
Degree $2$
Conductor $430$
Sign $-0.596 - 0.802i$
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  − 2-s + (1.69 + 2.93i)3-s + 4-s + (0.5 + 0.866i)5-s + (−1.69 − 2.93i)6-s + (0.736 − 1.27i)7-s − 8-s + (−4.23 + 7.34i)9-s + (−0.5 − 0.866i)10-s − 2.07·11-s + (1.69 + 2.93i)12-s + (−2.30 + 3.99i)13-s + (−0.736 + 1.27i)14-s + (−1.69 + 2.93i)15-s + 16-s + (3.97 − 6.88i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.978 + 1.69i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.691 − 1.19i)6-s + (0.278 − 0.482i)7-s − 0.353·8-s + (−1.41 + 2.44i)9-s + (−0.158 − 0.273i)10-s − 0.626·11-s + (0.489 + 0.846i)12-s + (−0.640 + 1.10i)13-s + (−0.196 + 0.341i)14-s + (−0.437 + 0.757i)15-s + 0.250·16-s + (0.964 − 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.596 - 0.802i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600476 + 1.19436i\)
\(L(\frac12)\) \(\approx\) \(0.600476 + 1.19436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-5.98 + 2.67i)T \)
good3 \( 1 + (-1.69 - 2.93i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.736 + 1.27i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 + (2.30 - 3.99i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.97 + 6.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.34 - 4.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.581 - 1.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.27 + 2.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.416 - 0.721i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.937 + 1.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.77T + 41T^{2} \)
47 \( 1 + 2.59T + 47T^{2} \)
53 \( 1 + (6.46 + 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.95T + 59T^{2} \)
61 \( 1 + (7.43 - 12.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.820 - 1.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.33 + 4.05i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.06 + 1.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.45 + 9.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.75 - 9.96i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.72 + 2.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−11.06598684438264497352943894808, −10.23356154440681379134137418385, −9.678511337430992314363491801655, −9.119009845774973895704775788547, −7.919568664498170358741478208164, −7.33960576575083600812680712570, −5.52230753770689283635686856700, −4.55896487154760253334163706098, −3.35289460544798395936692915178, −2.36290178484254430412102330027, 0.988775881180844165727860051967, 2.23779833599480221305861980181, 3.15394659107393319413115467820, 5.46494775303675038609788959304, 6.37902267887317964281372095900, 7.60514902572441410167546470836, 7.978413020716811365763511148352, 8.746443134430572471401103898623, 9.623991270234256965662141201979, 10.80321964391050994015990069679

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