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Properties

Label 2-750-5.2-c2-0-5
Degree $2$
Conductor $750$
Sign $-0.707 - 0.707i$
Analytic cond. $20.4360$
Root an. cond. $4.52062$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + 2.44·6-s + (2.69 + 2.69i)7-s + (2 − 2i)8-s − 2.99i·9-s + 1.76·11-s + (−2.44 − 2.44i)12-s + (−13.2 + 13.2i)13-s − 5.38i·14-s − 4·16-s + (13.3 + 13.3i)17-s + (−2.99 + 2.99i)18-s − 13.5i·19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + 0.408·6-s + (0.384 + 0.384i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + 0.160·11-s + (−0.204 − 0.204i)12-s + (−1.02 + 1.02i)13-s − 0.384i·14-s − 0.250·16-s + (0.786 + 0.786i)17-s + (−0.166 + 0.166i)18-s − 0.710i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(20.4360\)
Root analytic conductor: \(4.52062\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5423674789\)
\(L(\frac12)\) \(\approx\) \(0.5423674789\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-2.69 - 2.69i)T + 49iT^{2} \)
11 \( 1 - 1.76T + 121T^{2} \)
13 \( 1 + (13.2 - 13.2i)T - 169iT^{2} \)
17 \( 1 + (-13.3 - 13.3i)T + 289iT^{2} \)
19 \( 1 + 13.5iT - 361T^{2} \)
23 \( 1 + (3.45 - 3.45i)T - 529iT^{2} \)
29 \( 1 + 7.20iT - 841T^{2} \)
31 \( 1 - 15.1T + 961T^{2} \)
37 \( 1 + (28.4 + 28.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 20.6T + 1.68e3T^{2} \)
43 \( 1 + (28.7 - 28.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-16.4 - 16.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (71.9 - 71.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 27.4iT - 3.48e3T^{2} \)
61 \( 1 + 78.6T + 3.72e3T^{2} \)
67 \( 1 + (32.0 + 32.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 111.T + 5.04e3T^{2} \)
73 \( 1 + (20.1 - 20.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 81.3iT - 6.24e3T^{2} \)
83 \( 1 + (42.2 - 42.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 118. iT - 7.92e3T^{2} \)
97 \( 1 + (-79.0 - 79.0i)T + 9.40e3iT^{2} \)
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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

−10.48318196650771037523004835261, −9.603483849987765703900006717655, −9.044412891752272446052193606911, −8.024525371809627599552913534381, −7.10721962200496787043823229364, −6.04312745096627864728951632864, −4.92081024670293992831198435105, −4.07625084966752251912580470276, −2.74891945487499234672059447094, −1.50528106900525856053301619564, 0.24267626632230975856896729472, 1.52958756984826269966364472888, 3.06605420508046229387175435488, 4.69066914183238295543031705910, 5.44088020512319457246494155440, 6.40451525518214393266887713757, 7.46985735131204634941538058007, 7.79484627696905651818403142833, 8.858318817826137112261420491165, 10.05141835654049899258865684210

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