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Properties

Label 2-435-145.102-c1-0-26
Degree $2$
Conductor $435$
Sign $0.291 + 0.956i$
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  + (0.771 + 0.615i)2-s + (−0.222 + 0.974i)3-s + (−0.228 − 1.00i)4-s + (−0.815 − 2.08i)5-s + (−0.771 + 0.615i)6-s + (1.37 − 0.863i)7-s + (1.29 − 2.69i)8-s + (−0.900 − 0.433i)9-s + (0.651 − 2.10i)10-s + (−5.82 − 2.03i)11-s + 1.02·12-s + (−5.79 − 2.02i)13-s + (1.59 + 0.179i)14-s + (2.21 − 0.332i)15-s + (0.803 − 0.386i)16-s − 2.92i·17-s + ⋯
L(s)  = 1  + (0.545 + 0.434i)2-s + (−0.128 + 0.562i)3-s + (−0.114 − 0.500i)4-s + (−0.364 − 0.931i)5-s + (−0.314 + 0.251i)6-s + (0.519 − 0.326i)7-s + (0.458 − 0.951i)8-s + (−0.300 − 0.144i)9-s + (0.205 − 0.666i)10-s + (−1.75 − 0.614i)11-s + 0.296·12-s + (−1.60 − 0.562i)13-s + (0.425 + 0.0478i)14-s + (0.570 − 0.0857i)15-s + (0.200 − 0.0967i)16-s − 0.709i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.291 + 0.956i$
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: \((2,\ 435,\ (\ :1/2),\ 0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03411 - 0.766086i\)
\(L(\frac12)\) \(\approx\) \(1.03411 - 0.766086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.815 + 2.08i)T \)
29 \( 1 + (-5.30 + 0.921i)T \)
good2 \( 1 + (-0.771 - 0.615i)T + (0.445 + 1.94i)T^{2} \)
7 \( 1 + (-1.37 + 0.863i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (5.82 + 2.03i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (5.79 + 2.02i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + 2.92iT - 17T^{2} \)
19 \( 1 + (-6.53 - 4.10i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (-4.87 - 0.548i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (-3.53 + 0.397i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (-2.38 - 1.14i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-6.46 + 6.46i)T - 41iT^{2} \)
43 \( 1 + (-1.53 - 1.92i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.93 - 1.41i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.164 - 1.45i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 + 3.52iT - 59T^{2} \)
61 \( 1 + (1.78 - 1.11i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (1.31 - 0.458i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (3.76 + 7.82i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.04 + 2.42i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-13.6 + 4.79i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (9.68 + 6.08i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (0.148 + 1.31i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (0.519 + 2.27i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.83244084796824049106822217795, −10.05514542489427444863111046319, −9.338825216629293929988907387172, −7.931321082972288657170515418900, −7.46578664083424056273028501442, −5.71006239234978544885141021607, −5.05284252413661372973177387895, −4.64107274797304816011116613810, −3.01560443046997242043215152038, −0.68997346888943033727931115025, 2.42759119732588358900164416114, 2.87462311344035866977763495861, 4.62796994435572354598863399040, 5.25729888835941771042615061434, 6.96195893145484114316155164526, 7.56721500244421573781740199486, 8.243694310082122905826922953822, 9.721052553461590035196597497527, 10.76080922409035618622069426537, 11.50828806585982427030022056887

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