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Properties

Label 2-2352-28.3-c1-0-19
Degree $2$
Conductor $2352$
Sign $0.832 + 0.553i$
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 + (−1.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 1.73i·15-s + (−3 − 1.73i)17-s + (1 + 1.73i)19-s + (−1 + 1.73i)25-s + 0.999·27-s + 9·29-s + (−2.5 + 4.33i)31-s + (1.5 − 0.866i)33-s + (−5 − 8.66i)37-s − 10.3i·41-s + 3.46i·43-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.670 + 0.387i)5-s + (−0.166 − 0.288i)9-s + (−0.452 − 0.261i)11-s − 0.447i·15-s + (−0.727 − 0.420i)17-s + (0.229 + 0.397i)19-s + (−0.200 + 0.346i)25-s + 0.192·27-s + 1.67·29-s + (−0.449 + 0.777i)31-s + (0.261 − 0.150i)33-s + (−0.821 − 1.42i)37-s − 1.62i·41-s + 0.528i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.832 + 0.553i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9232449533\)
\(L(\frac12)\) \(\approx\) \(0.9232449533\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (3 + 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12 - 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-3 + 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 19.0iT - 97T^{2} \)
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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

−8.756327652377679620402074960724, −8.319630403852854032092117916114, −7.19806901918613571728673618656, −6.78081907778318147192799522950, −5.59088040331932855406511781194, −5.01518628788412728756419260601, −3.94338253428313960853257266351, −3.33540839094935822959404921408, −2.17958298347073021326386965882, −0.42031193331433056464883398447, 0.907964854197658979566790830164, 2.20278394035975240260261568231, 3.24472425020464416750413971669, 4.48185062284219992318060024646, 4.89949716431670803921462181651, 6.11619950060724635160308281746, 6.67797606702674631144498205572, 7.66034326653501779110874636638, 8.155156944134884325638296327909, 8.870358082145670571259780866552

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