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Properties

Label 2-435-145.102-c1-0-15
Degree $2$
Conductor $435$
Sign $0.982 - 0.187i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
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  + (1.06 + 0.852i)2-s + (0.222 − 0.974i)3-s + (−0.0286 − 0.125i)4-s + (−0.0950 + 2.23i)5-s + (1.06 − 0.852i)6-s + (1.16 − 0.729i)7-s + (1.26 − 2.62i)8-s + (−0.900 − 0.433i)9-s + (−2.00 + 2.30i)10-s + (0.223 + 0.0780i)11-s − 0.128·12-s + (5.64 + 1.97i)13-s + (1.86 + 0.210i)14-s + (2.15 + 0.589i)15-s + (3.35 − 1.61i)16-s − 3.11i·17-s + ⋯
L(s)  = 1  + (0.756 + 0.603i)2-s + (0.128 − 0.562i)3-s + (−0.0143 − 0.0628i)4-s + (−0.0425 + 0.999i)5-s + (0.436 − 0.348i)6-s + (0.439 − 0.275i)7-s + (0.446 − 0.927i)8-s + (−0.300 − 0.144i)9-s + (−0.634 + 0.729i)10-s + (0.0672 + 0.0235i)11-s − 0.0372·12-s + (1.56 + 0.547i)13-s + (0.498 + 0.0561i)14-s + (0.556 + 0.152i)15-s + (0.839 − 0.404i)16-s − 0.755i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.982 - 0.187i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.982 - 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24205 + 0.211774i\)
\(L(\frac12)\) \(\approx\) \(2.24205 + 0.211774i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.0950 - 2.23i)T \)
29 \( 1 + (2.48 + 4.77i)T \)
good2 \( 1 + (-1.06 - 0.852i)T + (0.445 + 1.94i)T^{2} \)
7 \( 1 + (-1.16 + 0.729i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (-0.223 - 0.0780i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (-5.64 - 1.97i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + 3.11iT - 17T^{2} \)
19 \( 1 + (-6.21 - 3.90i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (7.64 + 0.861i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (2.93 - 0.330i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (-7.36 - 3.54i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (3.76 - 3.76i)T - 41iT^{2} \)
43 \( 1 + (3.94 + 4.94i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.769 - 0.370i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.40 - 12.4i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + (9.02 - 5.66i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (6.30 - 2.20i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (4.47 + 9.29i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (7.43 - 5.93i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-3.29 + 1.15i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (-3.24 - 2.03i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (0.661 + 5.86i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (0.266 + 1.16i)T + (-87.3 + 42.0i)T^{2} \)
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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

−11.35002790011831353994905566421, −10.32427079973934705117740625280, −9.444170078643841959109494857404, −7.961961052709199103193201424740, −7.35766192837566446219312410114, −6.27560367663425711956140416815, −5.82926558231295929212157272047, −4.30546129645907424958094847566, −3.34211335220211972123882691658, −1.53542814045697036566576054013, 1.67638827920202964497579976947, 3.33754734188529746918107589705, 4.10314931355441950019968545573, 5.15381141923909915956677867919, 5.84064116056681920246755309336, 7.79940073058182357555586783921, 8.450742088120347775721240724663, 9.231822748540002405605874877607, 10.44764933213365734421408544246, 11.39727438470792202335221331564

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