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Properties

Label 2-750-25.11-c1-0-15
Degree $2$
Conductor $750$
Sign $-0.992 + 0.125i$
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 − 2.61·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−2.92 + 2.12i)11-s + (−0.809 − 0.587i)12-s + (−5.23 − 3.80i)13-s + (−2.11 + 1.53i)14-s + (−0.809 − 0.587i)16-s + (−0.381 − 1.17i)17-s − 0.999·18-s + (−1.76 − 5.42i)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 − 0.989·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.882 + 0.641i)11-s + (−0.233 − 0.169i)12-s + (−1.45 − 1.05i)13-s + (−0.566 + 0.411i)14-s + (−0.202 − 0.146i)16-s + (−0.0926 − 0.285i)17-s − 0.235·18-s + (−0.404 − 1.24i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.992 + 0.125i$
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: \((2,\ 750,\ (\ :1/2),\ -0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0715546 - 1.13732i\)
\(L(\frac12)\) \(\approx\) \(0.0715546 - 1.13732i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 + (2.92 - 2.12i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (5.23 + 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.381 + 1.17i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.76 + 5.42i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.61 + 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.61 + 8.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.04 - 6.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.47 - 4.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.61 + 3.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 + (0.527 - 1.62i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.645 + 1.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.92 - 2.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.472 - 1.45i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.70 + 5.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.85 + 2.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.73 - 5.34i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.663 + 2.04i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.85 - 2.07i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.04 + 3.21i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.03226753416087475894870418119, −9.330021885676637322082569813837, −8.091100462934196733930448758925, −7.17260352575813036295791442362, −6.49425275758425681181527509513, −5.29860341876129680208873605535, −4.52585200695713336751574617863, −2.84261648531462472563463632452, −2.57965309309323187956058703103, −0.43229705996656408973055262755, 2.44531875436386944784235279293, 3.41188823522727756056641498299, 4.42043738247822524402165565317, 5.38684483238586062945036851084, 6.26758317419519831544758365727, 7.21968803164111046343886818112, 8.096756801016053967575692577108, 9.146437150064208907932033039893, 9.835704270652937440418678551319, 10.69190738027145850542457980325

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