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Properties

Label 2-750-25.19-c1-0-2
Degree $2$
Conductor $750$
Sign $0.320 - 0.947i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (−0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.809 + 0.587i)6-s + 0.381i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.427 + 1.31i)11-s + (−0.951 − 0.309i)12-s + (−2.35 − 0.763i)13-s + (−0.118 − 0.363i)14-s + (0.309 − 0.951i)16-s + (−1.90 + 2.61i)17-s − i·18-s + (6.23 + 4.53i)19-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (−0.339 − 0.467i)3-s + (0.404 − 0.293i)4-s + (0.330 + 0.239i)6-s + 0.144i·7-s + (−0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + (0.128 + 0.396i)11-s + (−0.274 − 0.0892i)12-s + (−0.652 − 0.211i)13-s + (−0.0315 − 0.0970i)14-s + (0.0772 − 0.237i)16-s + (−0.461 + 0.634i)17-s − 0.235i·18-s + (1.43 + 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617941 + 0.443361i\)
\(L(\frac12)\) \(\approx\) \(0.617941 + 0.443361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 0.381iT - 7T^{2} \)
11 \( 1 + (-0.427 - 1.31i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.35 + 0.763i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.90 - 2.61i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.23 - 4.53i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (4.25 - 1.38i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.381 - 0.277i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.54 + 2.57i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-7.60 - 2.47i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.38 - 7.33i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.70iT - 43T^{2} \)
47 \( 1 + (-6.88 - 9.47i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.34 - 7.35i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.427 + 1.31i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.23 - 6.88i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.15 - 8.47i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (11.7 - 8.50i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.8 + 3.85i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.73 - 1.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.20 - 7.16i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.85 + 11.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.30 - 4.54i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.30903395225495772425880714691, −9.751699946501010633984178061666, −8.783134409390440812852587290991, −7.72641471996804635348001775701, −7.33745618360385437585540296403, −6.12299020481427148250888297644, −5.51415110940174963113593442299, −4.14951739035342314462502899600, −2.58385358872601478500986456514, −1.33086123120690473246184308799, 0.54293994500568535262353463834, 2.35431053451518994067959890049, 3.55557894929212799959039579398, 4.73624871228894581022202936713, 5.71522325168745902559236752068, 6.89877307996148447266266693493, 7.54967159978034433024082708547, 8.757219862636812357741562187000, 9.391042635593303076512285879678, 10.10810871842704556614569983567

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