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Properties

Label 2-430-43.11-c1-0-8
Degree $2$
Conductor $430$
Sign $0.603 + 0.797i$
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 + (1.14 − 1.43i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s − 1.82·6-s + 4.80·7-s + (0.900 − 0.433i)8-s + (−0.0773 − 0.338i)9-s + (0.900 + 0.433i)10-s + (1.16 + 5.10i)11-s + (1.14 + 1.43i)12-s + (3.55 − 1.71i)13-s + (−2.99 − 3.75i)14-s + (−0.407 + 1.78i)15-s + (−0.900 − 0.433i)16-s + (−3.09 − 1.49i)17-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (0.658 − 0.825i)3-s + (−0.111 + 0.487i)4-s + (−0.402 + 0.194i)5-s − 0.746·6-s + 1.81·7-s + (0.318 − 0.153i)8-s + (−0.0257 − 0.112i)9-s + (0.284 + 0.137i)10-s + (0.351 + 1.53i)11-s + (0.329 + 0.412i)12-s + (0.984 − 0.474i)13-s + (−0.800 − 1.00i)14-s + (−0.105 + 0.460i)15-s + (−0.225 − 0.108i)16-s + (−0.751 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.603 + 0.797i$
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: \((2,\ 430,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39682 - 0.695069i\)
\(L(\frac12)\) \(\approx\) \(1.39682 - 0.695069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 + 6.15i)T \)
good3 \( 1 + (-1.14 + 1.43i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 - 4.80T + 7T^{2} \)
11 \( 1 + (-1.16 - 5.10i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-3.55 + 1.71i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (3.09 + 1.49i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + (0.793 - 3.47i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (1.48 + 6.51i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-0.375 - 0.471i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (2.24 + 2.81i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + 7.64T + 37T^{2} \)
41 \( 1 + (5.28 + 6.62i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (2.39 - 10.4i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-5.00 - 2.40i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (7.64 + 3.68i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-7.07 + 8.87i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (1.06 - 4.64i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.106 + 0.467i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (5.59 - 2.69i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 0.197T + 79T^{2} \)
83 \( 1 + (-8.50 + 10.6i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (7.76 - 9.73i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-1.36 - 5.96i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.94907886644951111696180894316, −10.38619357137974935204963231406, −8.895217500962253467711446649380, −8.262960965670631193492553431173, −7.63759535223161569241072205254, −6.79737985673530218767590993669, −4.95755697335993728221675460564, −3.99076820374091713751415341699, −2.26660974662621342625808691374, −1.56682182178584585265538012384, 1.46184988280325867241812526689, 3.51466633577397085825686453347, 4.44678589347408812215095371233, 5.47042660002620741850623610569, 6.73593495081125828294577305496, 8.048465478559283369252749416957, 8.681870321759523081548266444184, 8.954216095117243493357326415519, 10.38295247609252804832211868097, 11.29276778118475032140291428533

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