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Properties

Label 2-435-29.24-c1-0-18
Degree $2$
Conductor $435$
Sign $-0.385 + 0.922i$
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  + (0.320 + 0.401i)2-s + (−0.222 − 0.974i)3-s + (0.386 − 1.69i)4-s + (−0.623 − 0.781i)5-s + (0.320 − 0.401i)6-s + (−0.169 − 0.744i)7-s + (1.72 − 0.832i)8-s + (−0.900 + 0.433i)9-s + (0.114 − 0.500i)10-s + (−0.761 − 0.366i)11-s − 1.73·12-s + (−5.11 − 2.46i)13-s + (0.244 − 0.306i)14-s + (−0.623 + 0.781i)15-s + (−2.24 − 1.07i)16-s + 5.13·17-s + ⋯
L(s)  = 1  + (0.226 + 0.283i)2-s + (−0.128 − 0.562i)3-s + (0.193 − 0.846i)4-s + (−0.278 − 0.349i)5-s + (0.130 − 0.163i)6-s + (−0.0641 − 0.281i)7-s + (0.611 − 0.294i)8-s + (−0.300 + 0.144i)9-s + (0.0361 − 0.158i)10-s + (−0.229 − 0.110i)11-s − 0.501·12-s + (−1.41 − 0.682i)13-s + (0.0653 − 0.0818i)14-s + (−0.160 + 0.201i)15-s + (−0.560 − 0.269i)16-s + 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.385 + 0.922i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697113 - 1.04673i\)
\(L(\frac12)\) \(\approx\) \(0.697113 - 1.04673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (4.76 + 2.50i)T \)
good2 \( 1 + (-0.320 - 0.401i)T + (-0.445 + 1.94i)T^{2} \)
7 \( 1 + (0.169 + 0.744i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (0.761 + 0.366i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (5.11 + 2.46i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 5.13T + 17T^{2} \)
19 \( 1 + (1.17 - 5.14i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-2.92 + 3.67i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (4.36 + 5.47i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (1.28 - 0.619i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + (-6.89 + 8.65i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-11.3 - 5.47i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-1.53 - 1.92i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 2.12T + 59T^{2} \)
61 \( 1 + (-2.91 - 12.7i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (3.74 - 1.80i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (3.60 + 1.73i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-6.28 + 7.88i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (10.6 - 5.12i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-2.12 + 9.29i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-6.92 - 8.68i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-0.648 + 2.84i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.70282255422355737702964389837, −10.14905670037249882439781900812, −9.091078110939347197965712297617, −7.59593531199727484542211493553, −7.41302140110402453617442451098, −5.88768014266234273492098076249, −5.40854209761474866728452516745, −4.10908063758663078927997713167, −2.36495539243129858037028279995, −0.74677239802277120144891295224, 2.42379180986097563359225948341, 3.39091626344950183146282798848, 4.50522185737734596632569340892, 5.46171894777573256501759152791, 7.09436134991411338693436941805, 7.52219647319416120396837413297, 8.862592395698007013205669547140, 9.613671981549427577024779389539, 10.77165093601684216222573648151, 11.40171805999219256436945394678

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