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Properties

Label 2-430-215.108-c1-0-17
Degree $2$
Conductor $430$
Sign $0.984 + 0.172i$
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  + (0.330 + 0.943i)2-s + (−0.142 + 0.408i)3-s + (−0.781 + 0.623i)4-s + (1.33 − 1.79i)5-s − 0.432·6-s + (−3.48 − 3.48i)7-s + (−0.846 − 0.532i)8-s + (2.19 + 1.75i)9-s + (2.13 + 0.664i)10-s + (2.65 − 3.32i)11-s + (−0.142 − 0.408i)12-s + (2.14 − 3.42i)13-s + (2.13 − 4.43i)14-s + (0.542 + 0.799i)15-s + (0.222 − 0.974i)16-s + (2.03 + 3.23i)17-s + ⋯
L(s)  = 1  + (0.233 + 0.667i)2-s + (−0.0824 + 0.235i)3-s + (−0.390 + 0.311i)4-s + (0.595 − 0.803i)5-s − 0.176·6-s + (−1.31 − 1.31i)7-s + (−0.299 − 0.188i)8-s + (0.733 + 0.584i)9-s + (0.675 + 0.209i)10-s + (0.799 − 1.00i)11-s + (−0.0412 − 0.117i)12-s + (0.596 − 0.948i)13-s + (0.571 − 1.18i)14-s + (0.140 + 0.206i)15-s + (0.0556 − 0.243i)16-s + (0.492 + 0.783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.984 + 0.172i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.984 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49327 - 0.129908i\)
\(L(\frac12)\) \(\approx\) \(1.49327 - 0.129908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.330 - 0.943i)T \)
5 \( 1 + (-1.33 + 1.79i)T \)
43 \( 1 + (6.27 + 1.90i)T \)
good3 \( 1 + (0.142 - 0.408i)T + (-2.34 - 1.87i)T^{2} \)
7 \( 1 + (3.48 + 3.48i)T + 7iT^{2} \)
11 \( 1 + (-2.65 + 3.32i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-2.14 + 3.42i)T + (-5.64 - 11.7i)T^{2} \)
17 \( 1 + (-2.03 - 3.23i)T + (-7.37 + 15.3i)T^{2} \)
19 \( 1 + (-0.925 - 1.16i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (1.57 - 0.177i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (0.351 + 0.169i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (1.11 + 0.537i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (3.68 + 3.68i)T + 37iT^{2} \)
41 \( 1 + (-7.64 - 3.68i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-12.1 - 1.36i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (0.472 - 0.296i)T + (22.9 - 47.7i)T^{2} \)
59 \( 1 + (10.8 + 2.48i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (-2.25 - 4.67i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-14.6 - 1.65i)T + (65.3 + 14.9i)T^{2} \)
71 \( 1 + (-4.13 + 3.29i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (7.64 + 4.80i)T + (31.6 + 65.7i)T^{2} \)
79 \( 1 - 6.63iT - 79T^{2} \)
83 \( 1 + (13.5 + 4.74i)T + (64.8 + 51.7i)T^{2} \)
89 \( 1 + (5.44 - 2.62i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-0.0457 - 0.406i)T + (-94.5 + 21.5i)T^{2} \)
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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

−10.83349200634933522702556845987, −10.13805266361750143241260820808, −9.379793082353198880293921862753, −8.321174521146966628800836143485, −7.36800323832581056819842305446, −6.25085339587003432774778695068, −5.63030512251653292597441395123, −4.19859757995891713870839335998, −3.50551068794103355878685496074, −0.999983247100098891021647455795, 1.79973379079509977791403538621, 2.93251540439754900262613585003, 4.02032182933506831172578695838, 5.59548473461816908337994937435, 6.51604823543098020588309788174, 7.04410293379069133238524924942, 9.033938046788348535048235525411, 9.511909737999074801784600974738, 10.06279076836227702744916570351, 11.40475726556829867473346614157

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