Properties

Label 2-756-63.38-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.802 - 0.597i$
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  + (0.842 − 1.45i)5-s + (−2.27 + 1.34i)7-s + (−3.38 + 1.95i)11-s + (−5.24 + 3.02i)13-s + (0.201 − 0.348i)17-s + (−0.145 + 0.0840i)19-s + (−7.69 − 4.44i)23-s + (1.07 + 1.86i)25-s + (6.15 + 3.55i)29-s + 6.28i·31-s + (0.0398 + 4.45i)35-s + (3.13 + 5.42i)37-s + (−1.64 − 2.85i)41-s + (1.80 − 3.12i)43-s − 8.76·47-s + ⋯
L(s)  = 1  + (0.376 − 0.652i)5-s + (−0.861 + 0.507i)7-s + (−1.01 + 0.588i)11-s + (−1.45 + 0.839i)13-s + (0.0488 − 0.0845i)17-s + (−0.0334 + 0.0192i)19-s + (−1.60 − 0.926i)23-s + (0.215 + 0.373i)25-s + (1.14 + 0.659i)29-s + 1.12i·31-s + (0.00673 + 0.753i)35-s + (0.514 + 0.891i)37-s + (−0.257 − 0.445i)41-s + (0.275 − 0.476i)43-s − 1.27·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.802 - 0.597i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.802 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128713 + 0.388312i\)
\(L(\frac12)\) \(\approx\) \(0.128713 + 0.388312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 - 1.34i)T \)
good5 \( 1 + (-0.842 + 1.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.38 - 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.24 - 3.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.201 + 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.145 - 0.0840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.69 + 4.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.15 - 3.55i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.28iT - 31T^{2} \)
37 \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.64 + 2.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.80 + 3.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.76T + 47T^{2} \)
53 \( 1 + (4.94 + 2.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.50T + 59T^{2} \)
61 \( 1 + 5.12iT - 61T^{2} \)
67 \( 1 + 5.91T + 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (6.05 + 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.20T + 79T^{2} \)
83 \( 1 + (0.181 - 0.314i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.38 - 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.508 + 0.293i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.27800998321305832188307333490, −9.925157779351912059987776867032, −9.063877176142105459239565972873, −8.225541744081198868787581554008, −7.13651124020991369721961721466, −6.32723477331352504983177638618, −5.17090765137387210383422973036, −4.58628301122662843978092980958, −2.97308003288481198993242853831, −1.98571416355605081414883535827, 0.18787846100842145357946259728, 2.44702421866417740972316816644, 3.17605430994653855354484977693, 4.50411036172446271230947621916, 5.74479669406666704057997728900, 6.36031932859214660112267991587, 7.57425270092455360752975420466, 7.966658911342822883040376886168, 9.484651450125111783223743805542, 10.11217220578781204938067612360

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