Properties

Label 2-2352-28.3-c1-0-28
Degree $2$
Conductor $2352$
Sign $0.769 + 0.638i$
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 + (3.72 − 2.14i)5-s + (−0.499 − 0.866i)9-s + (4.38 + 2.53i)11-s + 3.37i·13-s − 4.29i·15-s + (−2.39 − 1.38i)17-s + (2.35 + 4.07i)19-s + (4.18 − 2.41i)23-s + (6.73 − 11.6i)25-s − 0.999·27-s + 2.46·29-s + (−2.84 + 4.93i)31-s + (4.38 − 2.53i)33-s + (1.16 + 2.02i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (1.66 − 0.960i)5-s + (−0.166 − 0.288i)9-s + (1.32 + 0.763i)11-s + 0.937i·13-s − 1.10i·15-s + (−0.581 − 0.335i)17-s + (0.539 + 0.934i)19-s + (0.872 − 0.503i)23-s + (1.34 − 2.33i)25-s − 0.192·27-s + 0.456·29-s + (−0.511 + 0.886i)31-s + (0.763 − 0.440i)33-s + (0.191 + 0.332i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.769 + 0.638i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.769 + 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.048334905\)
\(L(\frac12)\) \(\approx\) \(3.048334905\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.72 + 2.14i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.38 - 2.53i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 + (2.39 + 1.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.35 - 4.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.18 + 2.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.46T + 29T^{2} \)
31 \( 1 + (2.84 - 4.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.16 - 2.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.14iT - 41T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + (-2.67 - 4.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.11 + 3.65i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.80 + 8.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.35 - 1.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.12 + 2.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (9.96 + 5.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.0 - 6.95i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.32T + 83T^{2} \)
89 \( 1 + (12.1 - 7.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−9.028755179397459469314631923666, −8.410038719646430007942386029236, −7.13470495544919996543422632069, −6.56208324860114691750373065127, −5.91320261542287442276444009362, −4.90705523421814756244396877682, −4.26110266530773649750885072942, −2.81516048288219000684706095013, −1.70705599785366863835706219948, −1.31972399209735990385348819206, 1.25784962803439150522974216508, 2.49597675735239482941890933082, 3.12070394462379750066552118507, 4.10542922030655550400056567397, 5.45057426589535035807279457890, 5.78858471877983614951246087296, 6.74161768189157832163294801320, 7.28897881099899927274740763290, 8.730938336361570650320154738942, 9.044547355955440374631626374862

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