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Properties

Label 2-750-5.4-c5-0-21
Degree $2$
Conductor $750$
Sign $-i$
Analytic cond. $120.287$
Root an. cond. $10.9675$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + 9i·3-s − 16·4-s + 36·6-s + 89.9i·7-s + 64i·8-s − 81·9-s − 360.·11-s − 144i·12-s + 616. i·13-s + 359.·14-s + 256·16-s − 1.26e3i·17-s + 324i·18-s + 2.59e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.693i·7-s + 0.353i·8-s − 0.333·9-s − 0.897·11-s − 0.288i·12-s + 1.01i·13-s + 0.490·14-s + 0.250·16-s − 1.06i·17-s + 0.235i·18-s + 1.64·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-i$
Analytic conductor: \(120.287\)
Root analytic conductor: \(10.9675\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :5/2),\ -i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.355885697\)
\(L(\frac12)\) \(\approx\) \(1.355885697\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 - 9iT \)
5 \( 1 \)
good7 \( 1 - 89.9iT - 1.68e4T^{2} \)
11 \( 1 + 360.T + 1.61e5T^{2} \)
13 \( 1 - 616. iT - 3.71e5T^{2} \)
17 \( 1 + 1.26e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.59e3T + 2.47e6T^{2} \)
23 \( 1 + 2.92e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.85e3T + 2.05e7T^{2} \)
31 \( 1 - 1.79e3T + 2.86e7T^{2} \)
37 \( 1 - 6.79e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.11e3T + 1.15e8T^{2} \)
43 \( 1 - 1.31e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.28e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.49e3iT - 4.18e8T^{2} \)
59 \( 1 - 6.22e3T + 7.14e8T^{2} \)
61 \( 1 + 3.12e4T + 8.44e8T^{2} \)
67 \( 1 - 3.99e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.35e4T + 1.80e9T^{2} \)
73 \( 1 - 1.19e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.39e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e5iT - 3.93e9T^{2} \)
89 \( 1 + 1.41e5T + 5.58e9T^{2} \)
97 \( 1 + 1.11e5iT - 8.58e9T^{2} \)
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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

−9.796670124681601177904538301611, −9.196586635885450022646057265405, −8.373087619002848160735311756824, −7.33625835696926983660627229430, −6.07774655075274479123226814407, −5.04622167654758696954462499057, −4.46744774994225185693864776976, −3.04308550966771693134668541099, −2.50618795783949398873776117377, −1.00898731511516196236038006084, 0.33551446271676391298902253303, 1.31217032226119584662627619885, 2.87357914633072568899640426042, 3.88455047863592764723087534593, 5.24793454549970592707873219927, 5.77535540230177132622143477095, 6.94343642214983562601460361399, 7.70058750167823906676773127615, 8.096546594018853533869364395870, 9.274958333251653347946384412018

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