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Properties

Label 2-2352-28.3-c1-0-9
Degree $2$
Conductor $2352$
Sign $0.0633 - 0.997i$
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 + (−2.12 + 1.22i)5-s + (−0.499 − 0.866i)9-s + (−2.12 − 1.22i)11-s − 4.89i·13-s − 2.44i·15-s + (2.12 + 1.22i)17-s + (1 + 1.73i)19-s + (6.36 − 3.67i)23-s + (0.499 − 0.866i)25-s + 0.999·27-s + 6·29-s + (−4 + 6.92i)31-s + (2.12 − 1.22i)33-s + (−2 − 3.46i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.948 + 0.547i)5-s + (−0.166 − 0.288i)9-s + (−0.639 − 0.369i)11-s − 1.35i·13-s − 0.632i·15-s + (0.514 + 0.297i)17-s + (0.229 + 0.397i)19-s + (1.32 − 0.766i)23-s + (0.0999 − 0.173i)25-s + 0.192·27-s + 1.11·29-s + (−0.718 + 1.24i)31-s + (0.369 − 0.213i)33-s + (−0.328 − 0.569i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.0633 - 0.997i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007893324\)
\(L(\frac12)\) \(\approx\) \(1.007893324\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 1.22i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.12 + 1.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 + (-2.12 - 1.22i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.36 + 3.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 + (12.7 + 7.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.48 - 4.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-10.6 + 6.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−9.062529674363141103971040563073, −8.343794959369085671894699170426, −7.65382108393328105765299681953, −6.98297981566335360142220377979, −5.89901052118215065652730917486, −5.24873053526900987029210012738, −4.34843140861437959207839131815, −3.25949690413410416749106224637, −2.92415285928886926240643224666, −0.913523190910416646209209140266, 0.47921110023694565691298562825, 1.74370731683984079919659315943, 2.94275885217016444434900147809, 4.07764025658447903415538303607, 4.81203620857356592707589930764, 5.54738202302543604243309939454, 6.68630256733579852883904472270, 7.31955119488803788602661608983, 7.87423301450517849954237422523, 8.789541558694064940193590388589

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