Properties

Label 2-75-15.14-c8-0-14
Degree $2$
Conductor $75$
Sign $-0.831 - 0.555i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23.7·2-s + (−10.0 + 80.3i)3-s + 309.·4-s + (−239. + 1.91e3i)6-s − 692. i·7-s + 1.28e3·8-s + (−6.35e3 − 1.61e3i)9-s + 1.58e4i·11-s + (−3.12e3 + 2.49e4i)12-s + 4.88e4i·13-s − 1.64e4i·14-s − 4.88e4·16-s − 4.10e4·17-s + (−1.51e5 − 3.85e4i)18-s + 1.08e5·19-s + ⋯
L(s)  = 1  + 1.48·2-s + (−0.124 + 0.992i)3-s + 1.21·4-s + (−0.184 + 1.47i)6-s − 0.288i·7-s + 0.313·8-s + (−0.969 − 0.246i)9-s + 1.08i·11-s + (−0.150 + 1.20i)12-s + 1.71i·13-s − 0.428i·14-s − 0.745·16-s − 0.491·17-s + (−1.44 − 0.367i)18-s + 0.830·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.831 - 0.555i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.830544 + 2.74121i\)
\(L(\frac12)\) \(\approx\) \(0.830544 + 2.74121i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (10.0 - 80.3i)T \)
5 \( 1 \)
good2 \( 1 - 23.7T + 256T^{2} \)
7 \( 1 + 692. iT - 5.76e6T^{2} \)
11 \( 1 - 1.58e4iT - 2.14e8T^{2} \)
13 \( 1 - 4.88e4iT - 8.15e8T^{2} \)
17 \( 1 + 4.10e4T + 6.97e9T^{2} \)
19 \( 1 - 1.08e5T + 1.69e10T^{2} \)
23 \( 1 + 4.33e5T + 7.83e10T^{2} \)
29 \( 1 + 3.77e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.08e5T + 8.52e11T^{2} \)
37 \( 1 - 2.31e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.15e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.42e6iT - 1.16e13T^{2} \)
47 \( 1 - 8.18e6T + 2.38e13T^{2} \)
53 \( 1 - 1.36e7T + 6.22e13T^{2} \)
59 \( 1 - 1.23e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.15e6T + 1.91e14T^{2} \)
67 \( 1 + 1.55e7iT - 4.06e14T^{2} \)
71 \( 1 - 2.52e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.29e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.49e7T + 1.51e15T^{2} \)
83 \( 1 - 2.12e7T + 2.25e15T^{2} \)
89 \( 1 + 4.87e7iT - 3.93e15T^{2} \)
97 \( 1 - 2.91e7iT - 7.83e15T^{2} \)
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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

−13.65057634731127482078328477164, −12.10527019060365460043832824444, −11.55913163501455297602640639565, −10.11681540150930394012219207134, −9.045253970968324008522488275478, −7.04141468692926111214784746305, −5.78645803696303031490545135241, −4.48403791730245091945893240256, −3.97090215075334987600467336190, −2.29321470528982742341628861178, 0.51032838882221603321094357112, 2.43405505801634162348970783625, 3.54388914763975056843929416475, 5.47180816101671427054773930215, 5.94838474432129268115143642332, 7.43804746626148133019362805465, 8.686346609224353932496533524899, 10.74614185345120626336555610439, 11.84155456802727056925207303419, 12.61819327056562377625597627082

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