Properties

Degree $2$
Conductor $2352$
Sign $0.900 - 0.435i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.292 − 0.507i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 5.41·13-s − 0.585·15-s + (3.12 + 5.40i)17-s + (−1.41 + 2.44i)19-s + (1.82 − 3.16i)23-s + (2.32 + 4.03i)25-s + 0.999·27-s − 1.17·29-s + (3.41 + 5.91i)31-s + (−0.999 + 1.73i)33-s + (2 − 3.46i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.130 − 0.226i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 1.50·13-s − 0.151·15-s + (0.757 + 1.31i)17-s + (−0.324 + 0.561i)19-s + (0.381 − 0.660i)23-s + (0.465 + 0.806i)25-s + 0.192·27-s − 0.217·29-s + (0.613 + 1.06i)31-s + (−0.174 + 0.301i)33-s + (0.328 − 0.569i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.900 - 0.435i$
Motivic weight: \(1\)
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270526915\)
\(L(\frac12)\) \(\approx\) \(1.270526915\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 3.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (-3.41 - 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.41 - 5.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.87 + 3.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (2.94 + 5.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.17 + 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (2.87 - 4.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.41T + 97T^{2} \)
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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

−8.928664395041748414792252184712, −8.192374514618714252386795993902, −7.52547543161031678091817333517, −6.74557186430433075315881230782, −5.84826869413173104508886197177, −5.25322973855090443721656628348, −4.31227277557701397878383861113, −3.15355701075482234813928683374, −2.16718439863959993082772461509, −0.993386080319218760693290486397, 0.53943860295613094382426913866, 2.33337366074546103979356501363, 2.98599309384768028519015982172, 4.30013901358592066771674641553, 4.94784064255828876783227819064, 5.59952470738727101180994847057, 6.72518904628038937448141126535, 7.31788330723041503582922430883, 8.067592093692999777163159058357, 9.194033316207359201968406008875

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