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Properties

Label 2-756-12.11-c1-0-19
Degree $2$
Conductor $756$
Sign $0.977 + 0.209i$
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  + (−1.40 − 0.148i)2-s + (1.95 + 0.418i)4-s + 1.05i·5-s − i·7-s + (−2.68 − 0.880i)8-s + (0.157 − 1.48i)10-s − 0.870·11-s + 2.65·13-s + (−0.148 + 1.40i)14-s + (3.64 + 1.63i)16-s − 1.50i·17-s − 3.94i·19-s + (−0.442 + 2.06i)20-s + (1.22 + 0.129i)22-s + 3.92·23-s + ⋯
L(s)  = 1  + (−0.994 − 0.105i)2-s + (0.977 + 0.209i)4-s + 0.472i·5-s − 0.377i·7-s + (−0.950 − 0.311i)8-s + (0.0497 − 0.469i)10-s − 0.262·11-s + 0.735·13-s + (−0.0398 + 0.375i)14-s + (0.912 + 0.409i)16-s − 0.364i·17-s − 0.904i·19-s + (−0.0989 + 0.461i)20-s + (0.260 + 0.0276i)22-s + 0.819·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.977 + 0.209i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.997436 - 0.105650i\)
\(L(\frac12)\) \(\approx\) \(0.997436 - 0.105650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.148i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 1.05iT - 5T^{2} \)
11 \( 1 + 0.870T + 11T^{2} \)
13 \( 1 - 2.65T + 13T^{2} \)
17 \( 1 + 1.50iT - 17T^{2} \)
19 \( 1 + 3.94iT - 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 - 0.285iT - 29T^{2} \)
31 \( 1 + 0.345iT - 31T^{2} \)
37 \( 1 - 3.07T + 37T^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 - 8.68iT - 43T^{2} \)
47 \( 1 - 7.32T + 47T^{2} \)
53 \( 1 + 14.4iT - 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 3.78T + 61T^{2} \)
67 \( 1 + 5.17iT - 67T^{2} \)
71 \( 1 - 7.45T + 71T^{2} \)
73 \( 1 - 5.01T + 73T^{2} \)
79 \( 1 - 6.95iT - 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 - 8.02T + 97T^{2} \)
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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

−10.29760025100194458815607759731, −9.472531528121286343026887154196, −8.668791644247447118854465657830, −7.79496454087046735780133791041, −6.93467526027143825079997204106, −6.30280006608003926874955788758, −4.94683861831492222097679955471, −3.46362685724613589368893532538, −2.52209588941626118874975566007, −0.932530466147160632636337300907, 1.06721624970908739231329528516, 2.40035514984417575133987801986, 3.73731789312569285325088729051, 5.26823844234670113804173924599, 6.04130681301840774915736566537, 7.05557517125857422973673881539, 7.997596102034555483876106112600, 8.762273244003200587322762965027, 9.270024159812278536653583999577, 10.44714301339828732989353586405

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