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Properties

Label 2-756-252.187-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.913 + 0.407i$
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  + (−1.34 − 0.439i)2-s + (1.61 + 1.18i)4-s + 2.88i·5-s + (−2.24 + 1.39i)7-s + (−1.64 − 2.29i)8-s + (1.26 − 3.87i)10-s − 2.30i·11-s + (−4.22 + 2.43i)13-s + (3.63 − 0.886i)14-s + (1.20 + 3.81i)16-s + (−3.54 + 2.04i)17-s + (0.308 − 0.534i)19-s + (−3.41 + 4.65i)20-s + (−1.01 + 3.09i)22-s − 8.41i·23-s + ⋯
L(s)  = 1  + (−0.950 − 0.311i)2-s + (0.806 + 0.591i)4-s + 1.28i·5-s + (−0.849 + 0.527i)7-s + (−0.582 − 0.812i)8-s + (0.401 − 1.22i)10-s − 0.693i·11-s + (−1.17 + 0.676i)13-s + (0.971 − 0.236i)14-s + (0.300 + 0.953i)16-s + (−0.859 + 0.496i)17-s + (0.0707 − 0.122i)19-s + (−0.762 + 1.04i)20-s + (−0.215 + 0.659i)22-s − 1.75i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.913 + 0.407i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.913 + 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00987394 - 0.0464167i\)
\(L(\frac12)\) \(\approx\) \(0.00987394 - 0.0464167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.439i)T \)
3 \( 1 \)
7 \( 1 + (2.24 - 1.39i)T \)
good5 \( 1 - 2.88iT - 5T^{2} \)
11 \( 1 + 2.30iT - 11T^{2} \)
13 \( 1 + (4.22 - 2.43i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.54 - 2.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.308 + 0.534i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.41iT - 23T^{2} \)
29 \( 1 + (-0.811 + 1.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.821 + 1.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.02 + 6.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.216 - 0.124i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.51 + 3.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.47 + 2.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.57 - 7.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.762 - 1.32i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.38 - 4.84i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.07 + 4.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (4.66 - 2.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.05 + 1.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.53 - 6.11i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (12.0 + 6.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 + 5.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.62897340245487445019184116540, −10.08273132652133507030107125978, −9.166885620783131925034124852515, −8.472823659889666603188107265922, −7.25912624541119918542355712729, −6.67781129940694056667374181076, −5.99114248240565178081874432951, −4.13448835005665230766689374321, −2.85320892319401305826914799497, −2.36077862634169003547987566102, 0.03175268471349705960564993057, 1.46397204738246835002388946190, 2.97831123744526555155116539165, 4.62851970695191972751603852128, 5.37886621382760553779936080529, 6.59009524733102352602475334714, 7.40597699729693654970063298650, 8.128662108393939133106089025053, 9.228164946964904903460745861783, 9.638254215495177776111958282676

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