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Properties

Label 2-1950-13.3-c1-0-43
Degree $2$
Conductor $1950$
Sign $-0.872 + 0.488i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s + (1 − 1.73i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s − 0.999·12-s + (3.5 − 0.866i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + 0.999·18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.204 − 0.353i)6-s + (0.377 − 0.654i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s − 0.288·12-s + (0.970 − 0.240i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + 0.235·18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8432620672\)
\(L(\frac12)\) \(\approx\) \(0.8432620672\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.5 + 0.866i)T \)
good7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.715927878463112715413151348196, −8.274442127281453992446882320501, −7.74853050479598642576672807887, −6.37676094692263901996284236351, −5.62554748738814150318699852792, −4.49192324638086014736300277949, −3.71382677890955821401430475414, −2.98608368215299611266035932527, −1.73483325226765150085499433309, −0.31905442401513903992629206416, 1.60552103111014607058052376755, 2.36785093839935018985393230343, 3.76051256825885755036106234753, 4.96379307788970275424519328362, 5.48689662705316968474346515922, 6.61878459451499097519724862605, 7.19750935368901669164601888926, 7.921729590370679079173151828655, 8.643717069149459106011447601934, 9.296293690340900925765819938554

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