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Properties

Label 2-750-5.3-c2-0-28
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 + (2.54 − 2.54i)7-s + (2 + 2i)8-s + 2.99i·9-s − 11.6·11-s + (2.44 − 2.44i)12-s + (−2.97 − 2.97i)13-s + 5.08i·14-s − 4·16-s + (−1.63 + 1.63i)17-s + (−2.99 − 2.99i)18-s + 2.25i·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 + (0.363 − 0.363i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s − 1.05·11-s + (0.204 − 0.204i)12-s + (−0.228 − 0.228i)13-s + 0.363i·14-s − 0.250·16-s + (−0.0963 + 0.0963i)17-s + (−0.166 − 0.166i)18-s + 0.118i·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: $\chi_{750} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01786447152\)
\(L(\frac12)\) \(\approx\) \(0.01786447152\)
\(L(2)\) 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 \( 1 + (-2.54 + 2.54i)T - 49iT^{2} \)
11 \( 1 + 11.6T + 121T^{2} \)
13 \( 1 + (2.97 + 2.97i)T + 169iT^{2} \)
17 \( 1 + (1.63 - 1.63i)T - 289iT^{2} \)
19 \( 1 - 2.25iT - 361T^{2} \)
23 \( 1 + (28.6 + 28.6i)T + 529iT^{2} \)
29 \( 1 - 49.9iT - 841T^{2} \)
31 \( 1 + 19.5T + 961T^{2} \)
37 \( 1 + (-10.2 + 10.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 43.8T + 1.68e3T^{2} \)
43 \( 1 + (10.4 + 10.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (46.2 - 46.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (40.8 + 40.8i)T + 2.80e3iT^{2} \)
59 \( 1 + 28.5iT - 3.48e3T^{2} \)
61 \( 1 + 14.8T + 3.72e3T^{2} \)
67 \( 1 + (-13.4 + 13.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 3.29T + 5.04e3T^{2} \)
73 \( 1 + (85.4 + 85.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 22.2iT - 6.24e3T^{2} \)
83 \( 1 + (21.6 + 21.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 113. iT - 7.92e3T^{2} \)
97 \( 1 + (-50.8 + 50.8i)T - 9.40e3iT^{2} \)
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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

−9.871267427380830141507109447229, −8.838610898015016748923825804091, −8.112037427692119208384253154460, −7.49068579358870436212086375159, −6.41139433704181797837832490991, −5.27974581551382222762090462719, −4.51338313724321799309567925399, −3.15957709053987339756528411066, −1.84075768504665590723421691405, −0.00641903091859303809232958145, 1.73343835930016730985477096873, 2.58777725606305323646351318231, 3.77416488248897557934038192202, 5.04412514006729014324238524990, 6.13053960347738532104167919143, 7.38667066435209301376055152620, 7.968943425584756668621627851282, 8.703284877669831325681766874937, 9.722068889290292010627052178330, 10.25575327096364257878461154194

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