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Properties

Label 2-756-63.25-c1-0-4
Degree $2$
Conductor $756$
Sign $0.483 + 0.875i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 + 2.18i)5-s + (0.527 − 2.59i)7-s + (−0.687 − 1.18i)11-s + (−2.80 − 4.84i)13-s + (2.69 − 4.66i)17-s + (2.44 + 4.23i)19-s + (2.08 − 3.61i)23-s + (−0.675 − 1.17i)25-s + (1.56 − 2.71i)29-s + 4.80·31-s + (4.99 + 4.41i)35-s + (−2.69 − 4.67i)37-s + (3.02 + 5.24i)41-s + (2.44 − 4.23i)43-s + 5.65·47-s + ⋯
L(s)  = 1  + (−0.563 + 0.976i)5-s + (0.199 − 0.979i)7-s + (−0.207 − 0.358i)11-s + (−0.776 − 1.34i)13-s + (0.653 − 1.13i)17-s + (0.561 + 0.972i)19-s + (0.435 − 0.753i)23-s + (−0.135 − 0.234i)25-s + (0.291 − 0.504i)29-s + 0.862·31-s + (0.844 + 0.746i)35-s + (−0.443 − 0.768i)37-s + (0.473 + 0.819i)41-s + (0.373 − 0.646i)43-s + 0.824·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03917 - 0.613494i\)
\(L(\frac12)\) \(\approx\) \(1.03917 - 0.613494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.527 + 2.59i)T \)
good5 \( 1 + (1.26 - 2.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.687 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.80 + 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.44 - 4.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.08 + 3.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.80T + 31T^{2} \)
37 \( 1 + (2.69 + 4.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.02 - 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.44 + 4.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + (-7.00 + 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 6.85T + 61T^{2} \)
67 \( 1 + 8.11T + 67T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + (-3.51 + 6.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.75T + 79T^{2} \)
83 \( 1 + (7.48 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.75 + 4.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.894 + 1.54i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.38824230084199742426151915721, −9.599640836995039086354579597449, −8.134261185080992416886988141887, −7.58671935295946533535218974831, −6.97380235744665588204675140332, −5.74756734556457499122125635523, −4.71663053334615776453273211671, −3.48278561281578931976143794825, −2.77796192102531242820933777191, −0.66359362749155985824201455230, 1.46476705718596269516980563428, 2.82627861936504235036103451594, 4.33409702507318894158703511449, 4.94065159060702897985244896814, 5.94836754944573727633166031204, 7.15110318051760981119911282722, 7.983396960555893577394250717221, 8.958811636507892060462503279855, 9.290992332486503599486937984542, 10.47645650035599529694698501505

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