↑

Properties

Label 2-936-104.99-c1-0-44
Degree $2$
Conductor $936$
Sign $0.957 - 0.289i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.00·4-s + (2.82 − 2.82i)5-s + (1 + i)7-s − 2.82i·8-s + (4.00 + 4.00i)10-s + (−1.41 + 1.41i)11-s + (−2 − 3i)13-s + (−1.41 + 1.41i)14-s + 4.00·16-s + 2.82i·17-s + (1 + i)19-s + (−5.65 + 5.65i)20-s + (−2.00 − 2.00i)22-s + 8.48·23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + (1.26 − 1.26i)5-s + (0.377 + 0.377i)7-s − 1.00i·8-s + (1.26 + 1.26i)10-s + (−0.426 + 0.426i)11-s + (−0.554 − 0.832i)13-s + (−0.377 + 0.377i)14-s + 1.00·16-s + 0.685i·17-s + (0.229 + 0.229i)19-s + (−1.26 + 1.26i)20-s + (−0.426 − 0.426i)22-s + 1.76·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77050 + 0.262156i\)
\(L(\frac12)\) \(\approx\) \(1.77050 + 0.262156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
13 \( 1 + (2 + 3i)T \)
good5 \( 1 + (-2.82 + 2.82i)T - 5iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (1.41 - 1.41i)T - 11iT^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + (-1 - i)T + 19iT^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + (-7 + 7i)T - 31iT^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + (-5.65 - 5.65i)T + 41iT^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + (-7.07 + 7.07i)T - 59iT^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + (9.89 - 9.89i)T - 71iT^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-9.89 - 9.89i)T + 83iT^{2} \)
89 \( 1 + (5.65 - 5.65i)T - 89iT^{2} \)
97 \( 1 + (-7 - 7i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.741516045703113180368052515879, −9.232557814808922384557791634634, −8.338447891430995512343439930492, −7.76331172524009456748299329849, −6.51168360143296946647637526215, −5.57609611317133313243438715147, −5.18649958556252205421469421587, −4.30622387659338216239599060642, −2.47145196150940810174359892889, −0.983196778336549892171486708245, 1.39910866146924081190762187500, 2.62707875287820081402961017309, 3.18179771749013310355984119111, 4.73945756716262267737653541427, 5.43136609808578778282734053478, 6.70072958320491420971095955299, 7.33279664106045445229271615226, 8.758204894563282903989929770427, 9.379467489961929225985018161623, 10.28063955721383013746549356219

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