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Properties

Label 2-2352-7.2-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 + (−0.414 − 0.717i)11-s − 1.41·13-s − 0.585·15-s + (−1.12 − 1.94i)17-s + (3.41 − 5.91i)19-s + (2.41 − 4.18i)23-s + (2.32 + 4.03i)25-s − 0.999·27-s + 8.48·29-s + (2.58 + 4.47i)31-s + (0.414 − 0.717i)33-s + (−0.828 + 1.43i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.130 + 0.226i)5-s + (−0.166 + 0.288i)9-s + (−0.124 − 0.216i)11-s − 0.392·13-s − 0.151·15-s + (−0.271 − 0.471i)17-s + (0.783 − 1.35i)19-s + (0.503 − 0.871i)23-s + (0.465 + 0.806i)25-s − 0.192·27-s + 1.57·29-s + (0.464 + 0.804i)31-s + (0.0721 − 0.124i)33-s + (−0.136 + 0.235i)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: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920204562\)
\(L(\frac12)\) \(\approx\) \(1.920204562\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.414 + 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (1.12 + 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.41 + 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.41 + 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + (-2.58 - 4.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.828 - 1.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-3.41 + 5.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.65 - 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.58 - 4.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.94 - 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.828T + 71T^{2} \)
73 \( 1 + (-5.53 - 9.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (-5.36 + 9.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−8.878321020265819061300812943292, −8.585197525581567466148170290903, −7.28554123437588842719545139074, −7.00734884964658761059128034707, −5.81995223188794844749664236469, −4.89278441993939403058011396713, −4.36646360424098482729497149772, −3.00778267007814502430851866402, −2.67263932808271957865276999013, −0.894114280058449976542923290038, 0.899765506938142028308698228459, 2.05681919804323685360089195547, 3.05971193608539679067963470247, 4.04006049474446831441921427451, 4.96156053222213495899242972913, 5.88985860945146900993556944987, 6.64094743856093348332262726760, 7.55455247153445323055763636183, 8.062774538429469927398071292459, 8.811215345484082955761354682002

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