Properties

Label 2-1260-140.27-c0-0-2
Degree $2$
Conductor $1260$
Sign $-0.229 + 0.973i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s − 5-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.707 + 0.707i)10-s − 1.41i·11-s + 1.00·14-s − 1.00·16-s + (1 − i)17-s − 1.41i·19-s + 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (0.707 − 0.707i)28-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s − 5-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.707 + 0.707i)10-s − 1.41i·11-s + 1.00·14-s − 1.00·16-s + (1 − i)17-s − 1.41i·19-s + 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (0.707 − 0.707i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337207566\)
\(L(\frac12)\) \(\approx\) \(1.337207566\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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

−9.683350743364153899078572234839, −8.890992832287390872452431187786, −8.142459364819612710232928877994, −7.16681271280985946135892853259, −6.05916156406297725524610447012, −5.19896686436612510562977372458, −4.52987198082840187326864923731, −3.31935060972493378996783704552, −2.72887767321862700831340790360, −1.01169715594222594540338677715, 1.90380789238496391701834953985, 3.66810844504495781289485995047, 4.01467021824935266083836555975, 4.97262316780492186515135190132, 5.86559465097898468830749126820, 7.07874672749192783409610740813, 7.61385779035667132795240750642, 8.030153026878363916304923860181, 9.068090236423037190830723535300, 10.26476029118101047528517954436

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