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Properties

Label 2-1950-13.3-c1-0-1
Degree $2$
Conductor $1950$
Sign $-0.597 + 0.802i$
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 + (−0.280 + 0.486i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−2.06 − 3.57i)11-s − 0.999·12-s + (−2.84 + 2.21i)13-s − 0.561·14-s + (−0.5 − 0.866i)16-s + (−1.56 + 2.70i)17-s − 0.999·18-s + (−0.280 + 0.486i)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 + (−0.106 + 0.183i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.621 − 1.07i)11-s − 0.288·12-s + (−0.788 + 0.615i)13-s − 0.150·14-s + (−0.125 − 0.216i)16-s + (−0.378 + 0.655i)17-s − 0.235·18-s + (−0.0644 + 0.111i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.597 + 0.802i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4893668326\)
\(L(\frac12)\) \(\approx\) \(0.4893668326\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.84 - 2.21i)T \)
good7 \( 1 + (0.280 - 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.280 - 0.486i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.34 - 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.21 + 2.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + (2.06 + 3.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.12 + 10.6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.219 + 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 8.56T + 53T^{2} \)
59 \( 1 + (3.21 - 5.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.12 + 1.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.56 + 11.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.36T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 7.12T + 83T^{2} \)
89 \( 1 + (-9.40 - 16.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.56 - 6.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.310511967254284628869574723528, −9.017628844667796205058150267516, −8.047153187859478726463728514700, −7.42791352214232481922900165719, −6.45486885986493508915774986965, −5.59506289509469387741373198455, −4.99096248255018624432697503198, −3.92698904711575687475767510934, −3.22044603385700993130486374991, −2.06246911768661774385854681975, 0.13747727961320575846342534922, 1.70171221590461482838926283432, 2.61496792528600889440224288682, 3.39058646748494261182910691169, 4.75433206836108902567565364406, 5.07665407213602374863475260071, 6.39815242730123495962426256673, 7.12954069653075658605678558675, 7.82726352974358342540108224358, 8.742311853920619968649172069625

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