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Properties

Label 2-750-125.104-c1-0-18
Degree $2$
Conductor $750$
Sign $-0.969 + 0.243i$
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  + (−0.248 − 0.968i)2-s + (−0.982 + 0.187i)3-s + (−0.876 + 0.481i)4-s + (−1.31 + 1.81i)5-s + (0.425 + 0.904i)6-s + (2.16 − 2.98i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (2.08 + 0.818i)10-s + (−5.53 + 1.42i)11-s + (0.770 − 0.637i)12-s + (1.86 + 4.70i)13-s + (−3.43 − 1.35i)14-s + (0.947 − 2.02i)15-s + (0.535 − 0.844i)16-s + (3.19 − 5.81i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.585 + 0.810i)5-s + (0.173 + 0.369i)6-s + (0.820 − 1.12i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (0.658 + 0.258i)10-s + (−1.67 + 0.428i)11-s + (0.222 − 0.184i)12-s + (0.516 + 1.30i)13-s + (−0.917 − 0.363i)14-s + (0.244 − 0.522i)15-s + (0.133 − 0.211i)16-s + (0.775 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0471735 - 0.382151i\)
\(L(\frac12)\) \(\approx\) \(0.0471735 - 0.382151i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.248 + 0.968i)T \)
3 \( 1 + (0.982 - 0.187i)T \)
5 \( 1 + (1.31 - 1.81i)T \)
good7 \( 1 + (-2.16 + 2.98i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (5.53 - 1.42i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (-1.86 - 4.70i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-3.19 + 5.81i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-0.739 + 3.87i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (6.72 + 0.422i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-2.43 - 0.306i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (5.69 + 3.12i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (1.97 + 1.25i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.473 + 7.51i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (5.28 - 1.71i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (6.75 - 7.19i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (8.22 + 3.86i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (7.00 + 8.46i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.395 + 6.28i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (0.664 + 5.26i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-6.97 - 6.54i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (-2.14 - 1.77i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (2.07 + 10.8i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (3.34 + 0.637i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-2.17 + 2.63i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (0.522 - 4.13i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.23244510516619607157533998808, −9.452491203138220081760876103856, −7.958094779152425528473629117723, −7.54770232476758482298522995285, −6.64752838889880245667384842418, −5.09927429915650967903776744295, −4.43352391289072181782794626724, −3.36697109023475203902137934317, −2.01332424509056310418404689007, −0.22715868319352560171136299492, 1.56013141558819045264280100572, 3.43347947572133453505000301249, 4.84929227831083935801424142319, 5.62830036046371374707831001273, 5.87469641159860006257808264677, 7.69787897452461025591933348551, 8.230048068290915940388430705731, 8.442855587341236966809584784229, 10.00675581383775699275957057788, 10.62151059168485291606075044400

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