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Properties

Label 2-2352-28.19-c1-0-17
Degree $2$
Conductor $2352$
Sign $0.654 + 0.756i$
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 + (−1.45 − 0.842i)5-s + (−0.499 + 0.866i)9-s + (3.05 − 1.76i)11-s + 2.93i·13-s + 1.68i·15-s + (−1.74 + 1.00i)17-s + (−0.848 + 1.47i)19-s + (1.38 + 0.799i)23-s + (−1.08 − 1.87i)25-s + 0.999·27-s + 7.94·29-s + (2.47 + 4.29i)31-s + (−3.05 − 1.76i)33-s + (5.23 − 9.06i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.652 − 0.376i)5-s + (−0.166 + 0.288i)9-s + (0.922 − 0.532i)11-s + 0.812i·13-s + 0.434i·15-s + (−0.422 + 0.243i)17-s + (−0.194 + 0.337i)19-s + (0.288 + 0.166i)23-s + (−0.216 − 0.374i)25-s + 0.192·27-s + 1.47·29-s + (0.444 + 0.770i)31-s + (−0.532 − 0.307i)33-s + (0.860 − 1.49i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.654 + 0.756i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.654 + 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.381702338\)
\(L(\frac12)\) \(\approx\) \(1.381702338\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 + 0.842i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.05 + 1.76i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.93iT - 13T^{2} \)
17 \( 1 + (1.74 - 1.00i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.848 - 1.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.38 - 0.799i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.94T + 29T^{2} \)
31 \( 1 + (-2.47 - 4.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.23 + 9.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.86iT - 41T^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 + (-3.35 + 5.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.46 + 2.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.87 - 6.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.9 + 6.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.47 + 0.850i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.13iT - 71T^{2} \)
73 \( 1 + (-2.10 + 1.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.749 + 0.432i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + (13.8 + 7.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.82iT - 97T^{2} \)
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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

−8.728657950997643805537749356411, −8.185815974543973420517657470889, −7.31941166319395704403104727756, −6.47132192248032653750445525511, −6.01744632020121439482711455460, −4.71247949642689278295435136828, −4.18913129373862380039150878371, −3.10342358224789113276755132185, −1.80838993296056781278868366965, −0.71114345742090429874452248209, 0.860067747887951788002737459050, 2.49577675897668260607628981543, 3.45449302820664915150347154915, 4.29921985810443113962320891352, 4.95013866001258802962875088491, 6.05700309249776431413180237570, 6.74785788310171299693514108013, 7.51132280329518087363200243980, 8.357637992344373968219306424575, 9.123023911989024147522298320314

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