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Properties

Label 2-1050-35.12-c1-0-2
Degree $2$
Conductor $1050$
Sign $-0.784 - 0.620i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
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  + (−0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s − i·6-s + (0.965 + 2.46i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.55 − 2.69i)11-s + (0.258 + 0.965i)12-s + (2.30 + 2.30i)13-s + (−1.57 − 2.12i)14-s + (0.500 − 0.866i)16-s + (1.10 + 0.295i)17-s + (0.965 + 0.258i)18-s + (−2.99 + 5.18i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s − 0.408i·6-s + (0.365 + 0.930i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.469 − 0.813i)11-s + (0.0747 + 0.278i)12-s + (0.639 + 0.639i)13-s + (−0.419 − 0.569i)14-s + (0.125 − 0.216i)16-s + (0.267 + 0.0716i)17-s + (0.227 + 0.0610i)18-s + (−0.687 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.784 - 0.620i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.784 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8238717343\)
\(L(\frac12)\) \(\approx\) \(0.8238717343\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.965 - 2.46i)T \)
good11 \( 1 + (1.55 + 2.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.30 - 2.30i)T + 13iT^{2} \)
17 \( 1 + (-1.10 - 0.295i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.99 - 5.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.295 - 1.10i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.39iT - 29T^{2} \)
31 \( 1 + (-3.68 + 2.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.55 - 1.48i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (6.44 - 6.44i)T - 43iT^{2} \)
47 \( 1 + (-1.10 - 4.11i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.78 - 0.746i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.117 - 0.203i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.38 - 4.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.60 - 13.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.74T + 71T^{2} \)
73 \( 1 + (-0.293 + 1.09i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.76 + 5.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.06 + 8.06i)T + 83iT^{2} \)
89 \( 1 + (9.18 - 15.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.740 + 0.740i)T - 97iT^{2} \)
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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

−10.25447863386620425348354800819, −9.296930263511296279379548206740, −8.534402500335069746496128647560, −8.172901855428278909893197181201, −6.85850957423310151550150698392, −5.88375107401887477361438921925, −5.37041256061391097553113157699, −4.04626696015335906205173031658, −2.89164447234959866231210675993, −1.56622395115109241635414996293, 0.47967691801338567979504145024, 1.75561956518681618516683034513, 2.94580505373849448665801604494, 4.26655433636607230863878626064, 5.29177580758347586113824750018, 6.58220637900048129100264991098, 7.11423081604293076364447683324, 8.035239148026691155715694919436, 8.522969441040113865565507797753, 9.760338351249215904819655679591

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