↑

Properties

Label 2-756-28.3-c1-0-45
Degree $2$
Conductor $756$
Sign $0.309 + 0.950i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (−0.271 + 1.38i)2-s + (−1.85 − 0.753i)4-s + (−0.945 + 0.545i)5-s + (2.64 − 0.133i)7-s + (1.54 − 2.36i)8-s + (−0.500 − 1.46i)10-s + (−4.97 − 2.86i)11-s + 1.80i·13-s + (−0.532 + 3.70i)14-s + (2.86 + 2.79i)16-s + (−4.79 − 2.76i)17-s + (−3.39 − 5.88i)19-s + (2.16 − 0.298i)20-s + (5.33 − 6.11i)22-s + (−1.99 + 1.14i)23-s + ⋯
L(s)  = 1  + (−0.192 + 0.981i)2-s + (−0.926 − 0.376i)4-s + (−0.422 + 0.244i)5-s + (0.998 − 0.0505i)7-s + (0.547 − 0.836i)8-s + (−0.158 − 0.461i)10-s + (−1.49 − 0.865i)11-s + 0.499i·13-s + (−0.142 + 0.989i)14-s + (0.715 + 0.698i)16-s + (−1.16 − 0.671i)17-s + (−0.779 − 1.34i)19-s + (0.483 − 0.0666i)20-s + (1.13 − 1.30i)22-s + (−0.415 + 0.239i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379751 - 0.275848i\)
\(L(\frac12)\) \(\approx\) \(0.379751 - 0.275848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.271 - 1.38i)T \)
3 \( 1 \)
7 \( 1 + (-2.64 + 0.133i)T \)
good5 \( 1 + (0.945 - 0.545i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.97 + 2.86i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.80iT - 13T^{2} \)
17 \( 1 + (4.79 + 2.76i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.39 + 5.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.99 - 1.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 + (-1.07 + 1.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.16 + 5.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.53iT - 41T^{2} \)
43 \( 1 + 7.52iT - 43T^{2} \)
47 \( 1 + (-1.77 - 3.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.57 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.56 - 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.69 + 3.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.51 - 0.872i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.88iT - 71T^{2} \)
73 \( 1 + (12.1 + 7.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.7 - 6.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + (-0.249 + 0.143i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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

−10.16732014532749564177513940247, −8.852686953269595768472642828708, −8.501081258476558951013206470709, −7.47958517617959304959198399637, −6.94118426838175143372306162955, −5.66653770301517942410821294768, −4.92808284003298656419709085472, −4.01060706023390710810730465952, −2.36423244522205282318538075810, −0.24872755583193459323778738418, 1.69893147198143219584341433697, 2.67383724025020716917446408071, 4.22455360164347646052262912976, 4.69526336022359003406864481460, 5.83885412506107005214982521313, 7.47765585062126109139750637941, 8.271844161252501641444970216230, 8.549151283467024239619532283056, 10.17532242667226569378034816072, 10.32916131863142558915611784620

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