↑

Properties

Label 2-1260-1260.319-c0-0-3
Degree $2$
Conductor $1260$
Sign $-0.400 + 0.916i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s − 0.999·12-s + 0.999·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s − 0.999·18-s + (−0.499 − 0.866i)20-s + 0.999·21-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s − 0.999·12-s + 0.999·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s − 0.999·18-s + (−0.499 − 0.866i)20-s + 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.400 + 0.916i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.746552466\)
\(L(\frac12)\) \(\approx\) \(1.746552466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−9.672026816753842575800472634733, −8.850130508116658125473080670859, −8.329319116788229000019469189790, −7.03475843117515900663027734047, −5.95100985378178132397221195550, −5.63444315280561529586028488552, −4.38243749978159051448568449607, −3.07074017650230868218650159233, −2.22644591345794300815023612081, −1.52662382932611715115673485702, 2.14245237210292350252710280574, 3.39293151592411561808389669340, 4.32010509593046336158152976171, 4.97635837280244271141031581626, 5.92654807042718380568895107303, 6.69369900147518386166570067170, 7.994670558406957060091102247179, 8.177744549812294514381730914513, 9.469849714049458013692771237260, 9.835317581853808439331342035573

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