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Properties

Label 2-435-29.23-c1-0-7
Degree $2$
Conductor $435$
Sign $0.874 - 0.485i$
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  + (1.12 − 1.41i)2-s + (−0.222 + 0.974i)3-s + (−0.280 − 1.22i)4-s + (−0.623 + 0.781i)5-s + (1.12 + 1.41i)6-s + (−0.863 + 3.78i)7-s + (1.20 + 0.580i)8-s + (−0.900 − 0.433i)9-s + (0.401 + 1.75i)10-s + (−2.71 + 1.30i)11-s + 1.25·12-s + (3.14 − 1.51i)13-s + (4.36 + 5.47i)14-s + (−0.623 − 0.781i)15-s + (4.44 − 2.14i)16-s − 0.506·17-s + ⋯
L(s)  = 1  + (0.795 − 0.997i)2-s + (−0.128 + 0.562i)3-s + (−0.140 − 0.613i)4-s + (−0.278 + 0.349i)5-s + (0.459 + 0.576i)6-s + (−0.326 + 1.43i)7-s + (0.426 + 0.205i)8-s + (−0.300 − 0.144i)9-s + (0.127 + 0.556i)10-s + (−0.819 + 0.394i)11-s + 0.363·12-s + (0.871 − 0.419i)13-s + (1.16 + 1.46i)14-s + (−0.160 − 0.201i)15-s + (1.11 − 0.535i)16-s − 0.122·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77989 + 0.461469i\)
\(L(\frac12)\) \(\approx\) \(1.77989 + 0.461469i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.01 + 4.46i)T \)
good2 \( 1 + (-1.12 + 1.41i)T + (-0.445 - 1.94i)T^{2} \)
7 \( 1 + (0.863 - 3.78i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (2.71 - 1.30i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.14 + 1.51i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 0.506T + 17T^{2} \)
19 \( 1 + (-1.65 - 7.24i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-1.20 - 1.50i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-6.24 + 7.83i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (4.19 + 2.01i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 3.48T + 41T^{2} \)
43 \( 1 + (-1.56 - 1.95i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.55 + 0.750i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-8.80 + 11.0i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 0.908T + 59T^{2} \)
61 \( 1 + (2.29 - 10.0i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-1.18 - 0.571i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (6.82 - 3.28i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-3.74 - 4.69i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-3.10 - 1.49i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (2.80 + 12.2i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-4.77 + 5.98i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (2.98 + 13.0i)T + (-87.3 + 42.0i)T^{2} \)
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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

−11.53545256224675462773424959286, −10.40149248767095620502953713050, −9.871315966456986289161227405336, −8.565005453784279097493190957219, −7.70751552352481694140860427410, −5.96147653520484150484828891961, −5.38717884243356105862966215016, −4.08149850411232654831425524303, −3.17598299161865842814873267098, −2.18746068449515652882381584646, 0.991563728316292742569864583807, 3.32131981478803806694359449494, 4.49724621762130877625333249533, 5.32365531782759963651533018932, 6.62475877641408248267351286517, 7.02329659975723641209982356367, 7.940451242908672444628285330827, 8.946850231689133910590693824944, 10.48748914644768952424709496437, 10.95791806622466553187746622508

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