↑

Properties

Label 2-1216-152.107-c1-0-16
Degree $2$
Conductor $1216$
Sign $0.0489 + 0.998i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (−3 + 1.73i)5-s + 2i·7-s + (−1 + 1.73i)9-s − 5.19·11-s + (−2 + 3.46i)13-s + (1.73 − 3i)15-s + (−3 − 5.19i)17-s + (4.33 + 0.5i)19-s + (−1 − 1.73i)21-s + (5.19 + 3i)23-s + (3.5 − 6.06i)25-s − 5i·27-s + (3 − 5.19i)29-s + 3.46·31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−1.34 + 0.774i)5-s + 0.755i·7-s + (−0.333 + 0.577i)9-s − 1.56·11-s + (−0.554 + 0.960i)13-s + (0.447 − 0.774i)15-s + (−0.727 − 1.26i)17-s + (0.993 + 0.114i)19-s + (−0.218 − 0.377i)21-s + (1.08 + 0.625i)23-s + (0.700 − 1.21i)25-s − 0.962i·27-s + (0.557 − 0.964i)29-s + 0.622·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0489 + 0.998i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.0489 + 0.998i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4.33 - 0.5i)T \)
good3 \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (10.5 - 6.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 - 3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.79 + 4.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.92 + 12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.46 - 6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.73T + 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \)
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

−9.627270333224181166061022258685, −8.643233285565000552072008995510, −7.66811583738342576276124484414, −7.32302503269987447709404054402, −6.17421609175915799090663488686, −4.99250643238685863328509286301, −4.67681990763283458848392361238, −3.08852116669487110032122849504, −2.53093700514597330789192335267, 0, 0.909885146323786746886071542109, 2.92439206493069812753360978391, 3.85463450283876190030054210511, 4.94435805433389926013613376925, 5.46675709717272358919402010849, 6.88357540077910711452515197336, 7.41980156580673756873665153129, 8.340041183721139417042885562547, 8.751965459075111202908554393786

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