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Properties

Label 2-750-25.11-c1-0-8
Degree $2$
Conductor $750$
Sign $0.970 + 0.242i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + 4.63·7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (2.05 − 1.49i)11-s + (−0.809 − 0.587i)12-s + (0.116 + 0.0846i)13-s + (−3.74 + 2.72i)14-s + (−0.809 − 0.587i)16-s + (2.31 + 7.12i)17-s + 0.999·18-s + (−2.08 − 6.41i)19-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.126 + 0.388i)6-s + 1.75·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (0.619 − 0.450i)11-s + (−0.233 − 0.169i)12-s + (0.0323 + 0.0234i)13-s + (−1.00 + 0.727i)14-s + (−0.202 − 0.146i)16-s + (0.561 + 1.72i)17-s + 0.235·18-s + (−0.477 − 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51029 - 0.185756i\)
\(L(\frac12)\) \(\approx\) \(1.51029 - 0.185756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 \)
good7 \( 1 - 4.63T + 7T^{2} \)
11 \( 1 + (-2.05 + 1.49i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.116 - 0.0846i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.31 - 7.12i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.08 + 6.41i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.35 - 0.985i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.696 - 2.14i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.310 - 0.954i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.0719 - 0.0523i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.48 + 1.80i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.02T + 43T^{2} \)
47 \( 1 + (-3.36 + 10.3i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.53 - 4.72i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.25 + 3.08i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.0 + 8.05i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.36 + 7.27i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.17 - 9.76i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.59 - 1.15i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.230 + 0.710i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.958 + 2.95i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.593 - 0.431i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.81 - 8.67i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.45450728259707061552804611418, −9.073142906064097104106268078731, −8.499802766556492857053196274339, −7.889692711407244470536777725445, −7.00844907038586633780354238319, −6.03665954623394064257228237423, −5.09456431107598817984373208103, −3.92098832794541791185736847375, −2.15007055811785485995582882279, −1.17562056151172202126716056166, 1.35484677019227857502566865287, 2.52409731690234515108136569963, 3.98838791655549327725069947319, 4.73346551023644984311910485098, 5.81641179230925406849907532461, 7.33615087447482723720433948847, 7.930764159355014591735181509612, 8.749506279664822466452823283557, 9.590116828581778191650850971821, 10.32572934727442351467110955669

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