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Properties

Label 2-750-125.104-c1-0-3
Degree $2$
Conductor $750$
Sign $-0.696 - 0.718i$
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 + (−2.01 − 0.975i)5-s + (−0.425 − 0.904i)6-s + (2.44 − 3.36i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (0.444 − 2.19i)10-s + (−3.71 + 0.952i)11-s + (0.770 − 0.637i)12-s + (1.92 + 4.85i)13-s + (3.86 + 1.52i)14-s + (2.15 + 0.580i)15-s + (0.535 − 0.844i)16-s + (−3.45 + 6.28i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.899 − 0.436i)5-s + (−0.173 − 0.369i)6-s + (0.923 − 1.27i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (0.140 − 0.693i)10-s + (−1.11 + 0.287i)11-s + (0.222 − 0.184i)12-s + (0.533 + 1.34i)13-s + (1.03 + 0.408i)14-s + (0.557 + 0.149i)15-s + (0.133 − 0.211i)16-s + (−0.838 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.696 - 0.718i$
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: \((2,\ 750,\ (\ :1/2),\ -0.696 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299722 + 0.707958i\)
\(L(\frac12)\) \(\approx\) \(0.299722 + 0.707958i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 + 0.975i)T \)
good7 \( 1 + (-2.44 + 3.36i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (3.71 - 0.952i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (-1.92 - 4.85i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (3.45 - 6.28i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.396 - 2.08i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-2.73 - 0.172i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (0.368 + 0.0465i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-7.58 - 4.17i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (3.48 + 2.21i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.404 - 6.43i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (11.9 - 3.87i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.39 + 2.54i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (2.84 + 1.33i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (7.23 + 8.75i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.380 - 6.04i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-1.09 - 8.65i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-10.9 - 10.3i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (2.15 + 1.78i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-2.85 - 14.9i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-11.0 - 2.10i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-0.121 + 0.146i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-0.416 + 3.30i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.85508780358662749366805294544, −9.931066005975288253572949811755, −8.478873811947135094945386712643, −8.151782402092761958719170384180, −7.12548447485061254513412290278, −6.46808987589017859520613446767, −5.04635174708300361164802415619, −4.47090218851899611844582549028, −3.77204014752348870100055018978, −1.42127539053049698048276873787, 0.42646873000220845280357198386, 2.40549158542528524748817541093, 3.18032146843740332610139383771, 4.84437040459612558013340748681, 5.16872711039560539680456370326, 6.35404536622341416561442018565, 7.65202285746486313993089039611, 8.291650346274893402923233806206, 9.162295657285905960060584830300, 10.55847277336070004685387338677

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