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Properties

Label 2-435-145.144-c1-0-27
Degree $2$
Conductor $435$
Sign $0.747 + 0.664i$
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  + 2·2-s − 3-s + 2·4-s + (2 − i)5-s − 2·6-s − 4i·7-s + 9-s + (4 − 2i)10-s + i·11-s − 2·12-s − 2i·13-s − 8i·14-s + (−2 + i)15-s − 4·16-s + 6·17-s + 2·18-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + (0.894 − 0.447i)5-s − 0.816·6-s − 1.51i·7-s + 0.333·9-s + (1.26 − 0.632i)10-s + 0.301i·11-s − 0.577·12-s − 0.554i·13-s − 2.13i·14-s + (−0.516 + 0.258i)15-s − 16-s + 1.45·17-s + 0.471·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.747 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45992 - 0.935260i\)
\(L(\frac12)\) \(\approx\) \(2.45992 - 0.935260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (-2 + i)T \)
29 \( 1 + (2 + 5i)T \)
good2 \( 1 - 2T + 2T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 9iT - 23T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 9iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 7iT - 83T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + 11T + 97T^{2} \)
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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

−11.31964077809128755766758793830, −10.07614116417574186181496133968, −9.777626087947438429350619065172, −7.969692709288310492536008623657, −7.00172992126185058755452452244, −5.87383952946041732115373603645, −5.34026607346045504613598208763, −4.27497789850775790366433240614, −3.33671717360574232261915277963, −1.36945748312626220732479902083, 2.22851511724058885799394735378, 3.18473962013872326817762539604, 4.74896060628684584323173366671, 5.54892767092728099639388658978, 6.10830894406878056202091453064, 6.95425659078922914311067993793, 8.696753834975057074824724346127, 9.476660412762067745707152239415, 10.66518176202523632624171352051, 11.53404899071342143673972193440

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