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Properties

Label 2-1050-35.33-c1-0-19
Degree $2$
Conductor $1050$
Sign $-0.770 + 0.637i$
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  + (−0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + i·6-s + (0.258 − 2.63i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (1.17 + 2.04i)11-s + (0.965 − 0.258i)12-s + (0.0968 − 0.0968i)13-s + (−2.61 + 0.431i)14-s + (0.500 − 0.866i)16-s + (0.833 − 3.11i)17-s + (0.258 − 0.965i)18-s + (0.434 − 0.752i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + 0.408i·6-s + (0.0978 − 0.995i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.355 + 0.615i)11-s + (0.278 − 0.0747i)12-s + (0.0268 − 0.0268i)13-s + (−0.697 + 0.115i)14-s + (0.125 − 0.216i)16-s + (0.202 − 0.754i)17-s + (0.0610 − 0.227i)18-s + (0.0996 − 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.770 + 0.637i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9621715749\)
\(L(\frac12)\) \(\approx\) \(0.9621715749\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.258 + 2.63i)T \)
good11 \( 1 + (-1.17 - 2.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0968 + 0.0968i)T - 13iT^{2} \)
17 \( 1 + (-0.833 + 3.11i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.434 + 0.752i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.11 + 0.833i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.08iT - 29T^{2} \)
31 \( 1 + (0.747 - 0.431i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.54 + 9.51i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.86iT - 41T^{2} \)
43 \( 1 + (-2.57 - 2.57i)T + 43iT^{2} \)
47 \( 1 + (-0.833 + 0.223i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.16 + 8.07i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.35 + 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.44 + 4.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.2 + 3.28i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 + (15.7 + 4.20i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.02 - 2.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.13 + 7.13i)T - 83iT^{2} \)
89 \( 1 + (-4.98 + 8.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.4 + 11.4i)T + 97iT^{2} \)
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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

−9.743607822275510740201210484266, −9.034608069853338934757194882711, −7.76971122542629473098738026073, −7.22767871816508794850023976656, −6.25529260502613927617225424951, −4.98814196903291833945961266554, −4.32278611835982891181756353279, −3.21331535925000857094979800848, −1.79282370505397805040633406141, −0.54575216813679515097828070339, 1.39344580644312816297141941292, 3.08805661345799418978857488317, 4.31471037264885991574249690674, 5.38985575638105807982257255531, 5.90833206527891466377255403317, 6.74440879160986860757020418466, 7.72576728434009462201850745240, 8.747119150200903365698062123353, 9.100951957090937710820617920070, 10.24401057892307012154124298928

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