Properties

Label 2-750-125.96-c1-0-11
Degree $2$
Conductor $750$
Sign $0.998 + 0.0632i$
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  + (−0.929 − 0.368i)2-s + (0.876 − 0.481i)3-s + (0.728 + 0.684i)4-s + (2.18 + 0.456i)5-s + (−0.992 + 0.125i)6-s + (2.24 + 1.63i)7-s + (−0.425 − 0.904i)8-s + (0.535 − 0.844i)9-s + (−1.86 − 1.22i)10-s + (3.95 + 1.56i)11-s + (0.968 + 0.248i)12-s + (1.60 − 2.53i)13-s + (−1.48 − 2.34i)14-s + (2.13 − 0.654i)15-s + (0.0627 + 0.998i)16-s + (−4.07 + 3.82i)17-s + ⋯
L(s)  = 1  + (−0.657 − 0.260i)2-s + (0.505 − 0.278i)3-s + (0.364 + 0.342i)4-s + (0.978 + 0.203i)5-s + (−0.405 + 0.0511i)6-s + (0.849 + 0.617i)7-s + (−0.150 − 0.319i)8-s + (0.178 − 0.281i)9-s + (−0.590 − 0.388i)10-s + (1.19 + 0.472i)11-s + (0.279 + 0.0717i)12-s + (0.445 − 0.702i)13-s + (−0.397 − 0.626i)14-s + (0.552 − 0.169i)15-s + (0.0156 + 0.249i)16-s + (−0.988 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.998 + 0.0632i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.998 + 0.0632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79497 - 0.0567876i\)
\(L(\frac12)\) \(\approx\) \(1.79497 - 0.0567876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.929 + 0.368i)T \)
3 \( 1 + (-0.876 + 0.481i)T \)
5 \( 1 + (-2.18 - 0.456i)T \)
good7 \( 1 + (-2.24 - 1.63i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-3.95 - 1.56i)T + (8.01 + 7.53i)T^{2} \)
13 \( 1 + (-1.60 + 2.53i)T + (-5.53 - 11.7i)T^{2} \)
17 \( 1 + (4.07 - 3.82i)T + (1.06 - 16.9i)T^{2} \)
19 \( 1 + (4.43 + 2.43i)T + (10.1 + 16.0i)T^{2} \)
23 \( 1 + (1.97 - 2.38i)T + (-4.30 - 22.5i)T^{2} \)
29 \( 1 + (-1.09 - 5.74i)T + (-26.9 + 10.6i)T^{2} \)
31 \( 1 + (-2.34 + 2.19i)T + (1.94 - 30.9i)T^{2} \)
37 \( 1 + (-0.277 - 4.40i)T + (-36.7 + 4.63i)T^{2} \)
41 \( 1 + (7.61 + 9.20i)T + (-7.68 + 40.2i)T^{2} \)
43 \( 1 + (1.00 + 3.10i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.40 - 2.98i)T + (-29.9 - 36.2i)T^{2} \)
53 \( 1 + (11.2 + 1.41i)T + (51.3 + 13.1i)T^{2} \)
59 \( 1 + (-8.00 - 2.05i)T + (51.7 + 28.4i)T^{2} \)
61 \( 1 + (-0.718 + 0.868i)T + (-11.4 - 59.9i)T^{2} \)
67 \( 1 + (-2.01 + 10.5i)T + (-62.2 - 24.6i)T^{2} \)
71 \( 1 + (-1.19 + 2.54i)T + (-45.2 - 54.7i)T^{2} \)
73 \( 1 + (-13.9 + 3.57i)T + (63.9 - 35.1i)T^{2} \)
79 \( 1 + (-5.91 + 3.25i)T + (42.3 - 66.7i)T^{2} \)
83 \( 1 + (13.9 + 7.67i)T + (44.4 + 70.0i)T^{2} \)
89 \( 1 + (-16.6 + 4.27i)T + (77.9 - 42.8i)T^{2} \)
97 \( 1 + (-2.14 - 11.2i)T + (-90.1 + 35.7i)T^{2} \)
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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

−10.32550828759609329090081880172, −9.285442173698476272798586012263, −8.747664355079757659302371684811, −8.094447078704207255902518654760, −6.78942424765419539936091196641, −6.25536856336068329153186655935, −4.93706477162175193513549548746, −3.58696008749613081875559543247, −2.18581611308541682592186426659, −1.59654045168538285414331214867, 1.30953382716110364465011199432, 2.31956135571254330061472083384, 4.02448833551121496821473640086, 4.86520416071135628131156652102, 6.29626786963753152758186259388, 6.74061134883057187511910914612, 8.121521561249326790902696583220, 8.664621792872957568436054087904, 9.436094964149205262769269514964, 10.11791224344339181850630719641

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