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Properties

Label 2-2352-28.3-c1-0-32
Degree $2$
Conductor $2352$
Sign $-0.414 + 0.909i$
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 + (0.521 − 0.300i)5-s + (−0.499 − 0.866i)9-s + (−0.141 − 0.0818i)11-s − 3.37i·13-s − 0.601i·15-s + (−1.84 − 1.06i)17-s + (0.476 + 0.825i)19-s + (6.05 − 3.49i)23-s + (−2.31 + 4.01i)25-s − 0.999·27-s − 1.28·29-s + (1.67 − 2.90i)31-s + (−0.141 + 0.0818i)33-s + (−2.58 − 4.47i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.233 − 0.134i)5-s + (−0.166 − 0.288i)9-s + (−0.0427 − 0.0246i)11-s − 0.937i·13-s − 0.155i·15-s + (−0.447 − 0.258i)17-s + (0.109 + 0.189i)19-s + (1.26 − 0.729i)23-s + (−0.463 + 0.803i)25-s − 0.192·27-s − 0.239·29-s + (0.301 − 0.521i)31-s + (−0.0246 + 0.0142i)33-s + (−0.424 − 0.735i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.414 + 0.909i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.414 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.647990652\)
\(L(\frac12)\) \(\approx\) \(1.647990652\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.521 + 0.300i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.141 + 0.0818i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + (1.84 + 1.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.476 - 0.825i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.05 + 3.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 + (-1.67 + 2.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.58 + 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 + (1.84 + 3.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.94 - 12.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.157 - 0.0907i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.87 + 4.54i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.05 + 3.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + (-12.8 + 7.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.08iT - 97T^{2} \)
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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

−8.891970171916635182868984105495, −7.84414381681701683309975257046, −7.34044831344160830021449720830, −6.44736033570519676589984948180, −5.62059873927880120801891907098, −4.87614819154692990884262720500, −3.69442798482422277593698358010, −2.82239694829746255556121451034, −1.82996737567467110439351400982, −0.53350886519229055282748513611, 1.47203709125877915517350876084, 2.59827876037185334931059433843, 3.49335823994874630332390614522, 4.48887133093485067758328120024, 5.09513131183298158114286709911, 6.22646856987758050565673299607, 6.82324957359571497297711181326, 7.78431578779459554781028451652, 8.580284472497845653152167355780, 9.277473493644721142222971692216

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