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Properties

Label 2-2352-7.4-c1-0-19
Degree $2$
Conductor $2352$
Sign $0.900 + 0.435i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (−0.5 + 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: $0.900 + 0.435i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
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

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

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