Properties

Label 2-2352-28.19-c1-0-13
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: $\chi_{2352} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.108269318\)
\(L(\frac12)\) \(\approx\) \(2.108269318\)
\(L(\frac{3}{2})\) 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

−9.313394464717530814495862460431, −8.395539702645185128657757603944, −7.79639381680352708535680533347, −6.76709204518085945495284690884, −6.05069780533584279307470182180, −5.19749676002646902385245885126, −4.53139977905019249359862998975, −3.05397835980079863949377736976, −2.79891924089608919773572298022, −1.40841232619491363115114189053, 0.69245663614210051488359043586, 2.01305091326423089361143897044, 2.58042292628804287151491408201, 3.94634767280332566531180311099, 4.94571184166375123241280904396, 5.58680340987879251229659355677, 6.76033634459101752882771030663, 6.89323688395092612823204537860, 8.299108883205657987306031533765, 8.707872492531038913327128192597

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