↑

Properties

Label 2-312-104.19-c1-0-10
Degree $2$
Conductor $312$
Sign $0.516 - 0.856i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
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  + (1.36 + 0.366i)2-s + (−0.5 + 0.866i)3-s + (1.73 + i)4-s + (−0.366 − 0.366i)5-s + (−1 + 0.999i)6-s + (2.36 + 0.633i)7-s + (1.99 + 2i)8-s + (−0.499 − 0.866i)9-s + (−0.366 − 0.633i)10-s + (−3.36 + 0.901i)11-s + (−1.73 + 0.999i)12-s + (3.5 + 0.866i)13-s + (3 + 1.73i)14-s + (0.5 − 0.133i)15-s + (1.99 + 3.46i)16-s + (0.232 − 0.133i)17-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.288 + 0.499i)3-s + (0.866 + 0.5i)4-s + (−0.163 − 0.163i)5-s + (−0.408 + 0.408i)6-s + (0.894 + 0.239i)7-s + (0.707 + 0.707i)8-s + (−0.166 − 0.288i)9-s + (−0.115 − 0.200i)10-s + (−1.01 + 0.271i)11-s + (−0.499 + 0.288i)12-s + (0.970 + 0.240i)13-s + (0.801 + 0.462i)14-s + (0.129 − 0.0345i)15-s + (0.499 + 0.866i)16-s + (0.0562 − 0.0324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.516 - 0.856i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.516 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88554 + 1.06439i\)
\(L(\frac12)\) \(\approx\) \(1.88554 + 1.06439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 + (0.366 + 0.366i)T + 5iT^{2} \)
7 \( 1 + (-2.36 - 0.633i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.36 - 0.901i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.232 + 0.133i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.09 + 1.09i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.366 + 0.633i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.59 + 1.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 - 4i)T + 31iT^{2} \)
37 \( 1 + (1.86 + 6.96i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.5 + 9.33i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.633 - 0.366i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.26 + 2.26i)T - 47iT^{2} \)
53 \( 1 + 8.66iT - 53T^{2} \)
59 \( 1 + (2.80 - 10.4i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.66 + 0.964i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.56 + 5.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.83 + 6.83i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.83 - 7.83i)T + 73iT^{2} \)
79 \( 1 + 1.07iT - 79T^{2} \)
83 \( 1 + (7.92 - 7.92i)T - 83iT^{2} \)
89 \( 1 + (12.5 - 3.36i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-15.2 - 4.09i)T + (84.0 + 48.5i)T^{2} \)
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

−11.90162952257856646647279287882, −10.98023261727202185254344345889, −10.42330698387598484509412948287, −8.724288762965444012560414214831, −8.008531530578253259793031593650, −6.75383827789170679945376645481, −5.61890095834128384723405248002, −4.80480854814687732993644536374, −3.85340499560761168925782122543, −2.24682492671853828606758675339, 1.53751776265104037776807544680, 3.05832744956174382926638588942, 4.44826226547852969822794463832, 5.48796365204231895919436538379, 6.40520609989305718624686698562, 7.58853288618617541058436681877, 8.334675012977016515514965554410, 10.10876519773513006669965256653, 11.09427041645605437370498318439, 11.33981058758933080540483830902

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