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Properties

Label 2-756-21.17-c1-0-10
Degree $2$
Conductor $756$
Sign $-0.832 + 0.553i$
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  + (−1.56 − 2.70i)5-s + (2.37 − 1.17i)7-s + (−3.23 − 1.87i)11-s − 2.93i·13-s + (−2.71 + 4.70i)17-s + (−6.07 + 3.50i)19-s + (1.44 − 0.832i)23-s + (−2.37 + 4.10i)25-s − 7.07i·29-s + (6.60 + 3.81i)31-s + (−6.87 − 4.57i)35-s + (−4.03 − 6.98i)37-s − 9.60·41-s − 5.74·43-s + (−1.56 − 2.70i)47-s + ⋯
L(s)  = 1  + (−0.697 − 1.20i)5-s + (0.895 − 0.444i)7-s + (−0.976 − 0.563i)11-s − 0.812i·13-s + (−0.658 + 1.14i)17-s + (−1.39 + 0.804i)19-s + (0.300 − 0.173i)23-s + (−0.474 + 0.821i)25-s − 1.31i·29-s + (1.18 + 0.685i)31-s + (−1.16 − 0.773i)35-s + (−0.663 − 1.14i)37-s − 1.49·41-s − 0.875·43-s + (−0.227 − 0.394i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246548 - 0.815807i\)
\(L(\frac12)\) \(\approx\) \(0.246548 - 0.815807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.37 + 1.17i)T \)
good5 \( 1 + (1.56 + 2.70i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.23 + 1.87i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + (2.71 - 4.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.07 - 3.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.44 + 0.832i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.07iT - 29T^{2} \)
31 \( 1 + (-6.60 - 3.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.03 + 6.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.60T + 41T^{2} \)
43 \( 1 + 5.74T + 43T^{2} \)
47 \( 1 + (1.56 + 2.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.11 - 1.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.20 + 3.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.57 - 1.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.90 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.33iT - 71T^{2} \)
73 \( 1 + (-2.46 - 1.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.57 + 4.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.39iT - 97T^{2} \)
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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

−10.32346863046652253074418917971, −8.718468112577308526393715186602, −8.253011493690421253984173700671, −7.86192383492110024244197898139, −6.40978246106315404237868329555, −5.29137926270043149282375334687, −4.56308626723682809819138454737, −3.66144192429461819279928942218, −1.93752046551127407429621743716, −0.41593176743631246662813349842, 2.13990056285615274306882285352, 3.00231879372333296607393338905, 4.45199583436243788504969816767, 5.09886811264860449438383940248, 6.70967932481633896710855870249, 7.03589858846628029993838196334, 8.121415315601305950164384454000, 8.819658983843808564116102776095, 10.01934255032701868893858105259, 10.83371969741841250282061369692

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