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Properties

Label 2-1050-5.3-c2-0-4
Degree $2$
Conductor $1050$
Sign $-0.973 - 0.229i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s − 2.44·6-s + (1.87 − 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s − 14.0·11-s + (2.44 − 2.44i)12-s + (6.03 + 6.03i)13-s + 3.74i·14-s − 4·16-s + (9.54 − 9.54i)17-s + (−2.99 − 2.99i)18-s + 21.6i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s − 0.408·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s − 1.28·11-s + (0.204 − 0.204i)12-s + (0.464 + 0.464i)13-s + 0.267i·14-s − 0.250·16-s + (0.561 − 0.561i)17-s + (−0.166 − 0.166i)18-s + 1.14i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8842877938\)
\(L(\frac12)\) \(\approx\) \(0.8842877938\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 + 14.0T + 121T^{2} \)
13 \( 1 + (-6.03 - 6.03i)T + 169iT^{2} \)
17 \( 1 + (-9.54 + 9.54i)T - 289iT^{2} \)
19 \( 1 - 21.6iT - 361T^{2} \)
23 \( 1 + (0.423 + 0.423i)T + 529iT^{2} \)
29 \( 1 + 11.2iT - 841T^{2} \)
31 \( 1 + 16.4T + 961T^{2} \)
37 \( 1 + (47.6 - 47.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 + (-46.7 - 46.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-20.3 + 20.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (-18.6 - 18.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 10.8T + 3.72e3T^{2} \)
67 \( 1 + (72.2 - 72.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 64.1T + 5.04e3T^{2} \)
73 \( 1 + (51.4 + 51.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 157. iT - 6.24e3T^{2} \)
83 \( 1 + (-76.9 - 76.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 37.6iT - 7.92e3T^{2} \)
97 \( 1 + (97.2 - 97.2i)T - 9.40e3iT^{2} \)
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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

−10.13066578103985183351987068282, −9.219190908743047616936939433973, −8.324062889991052415391448778778, −7.81037315095978809152984192695, −6.96083080284652814722929659536, −5.78572112084250117223703406789, −5.05396985036773621020524812198, −3.97517641505770372275475293634, −2.78963337018509636923494730274, −1.44799408187290145096079902108, 0.30264618496058805028190434627, 1.73787168884313443880635101389, 2.72628163168141357247816899852, 3.62576497981924301619271650563, 5.00130153579642093378864597411, 5.89028646395753593741365088073, 7.20404070656300960427667647648, 7.73915464066676006710952421290, 8.659445906024901210698576263657, 9.105364390210917339257789992733

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