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Properties

Label 2-1950-65.29-c1-0-25
Degree $2$
Conductor $1950$
Sign $-0.0342 + 0.999i$
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  + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + (−1.73 − i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1 − 1.73i)11-s + 0.999i·12-s + (2.59 + 2.5i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (4.33 + 2.5i)17-s − 0.999i·18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s + (−0.654 − 0.377i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.301 − 0.522i)11-s + 0.288i·12-s + (0.720 + 0.693i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (1.05 + 0.606i)17-s − 0.235i·18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0342 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0342 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0342 + 0.999i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.0342 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722619606\)
\(L(\frac12)\) \(\approx\) \(1.722619606\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-2.59 - 2.5i)T \)
good7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.33 - 2.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-9.52 + 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.66 + 5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.73 + i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.215746470781293505810015751751, −8.214066965552417896858721520709, −7.22432761620875432977219670584, −6.29637648494992533254269217621, −5.84318407051633069221411325609, −4.89775483452651950602571463866, −3.83048036204672428938971351882, −3.42981113604179270070234333560, −1.99848931585558765634756222222, −0.58527917790090619339315446663, 1.27614535732585334893225109974, 2.82617662725035052444046728681, 3.45704097557826972763104011884, 4.80150889424386822749895787386, 5.40158849775040088339015123524, 6.12682043013550900797513725545, 6.95431935113321805623160794238, 7.57250309441479910301852639783, 8.446923157207888788898746696663, 9.428550129888476964603605920622

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