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Properties

Label 2-2352-28.19-c1-0-37
Degree $2$
Conductor $2352$
Sign $-0.895 - 0.444i$
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 − 1.73i)5-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s − 1.73i·13-s + 3.46i·15-s + (2.5 − 4.33i)19-s + (−6 − 3.46i)23-s + (3.5 + 6.06i)25-s + 0.999·27-s + (−2.5 − 4.33i)31-s + (−3 − 1.73i)33-s + (5.5 − 9.52i)37-s + (−1.49 + 0.866i)39-s − 3.46i·41-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−1.34 − 0.774i)5-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s − 0.480i·13-s + 0.894i·15-s + (0.573 − 0.993i)19-s + (−1.25 − 0.722i)23-s + (0.700 + 1.21i)25-s + 0.192·27-s + (−0.449 − 0.777i)31-s + (−0.522 − 0.301i)33-s + (0.904 − 1.56i)37-s + (−0.240 + 0.138i)39-s − 0.541i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.895 - 0.444i$
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: \((2,\ 2352,\ (\ :1/2),\ -0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5474726415\)
\(L(\frac12)\) \(\approx\) \(0.5474726415\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6 + 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.5 + 6.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−8.449942892896759581205198928803, −7.78290243696478170870578388742, −7.17041631603365486434514503538, −6.20002311406581522914434435362, −5.42210625503084125942408110976, −4.36821327415526520552638126455, −3.84715448270943117777029326617, −2.63638186442779382525477860321, −1.12382345083544906122864893634, −0.22836289856137411672963545183, 1.59631233487813131159050231603, 3.12720229769520223265923479699, 3.84558346434773473541643891978, 4.35880064904406542696236399975, 5.47641920194200923920322113295, 6.48775595233884848843344908526, 7.06711350784399495827442739540, 7.88629172481279021040580104344, 8.528595555660986011144992333201, 9.671698580830243451694689247146

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