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Properties

Label 2-1950-13.4-c1-0-30
Degree $2$
Conductor $1950$
Sign $0.947 - 0.319i$
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 + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (2.42 − 1.40i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.515 − 0.297i)11-s − 0.999·12-s + (3.43 − 1.10i)13-s + 2.80·14-s + (−0.5 + 0.866i)16-s + (−2.87 − 4.98i)17-s − 0.999i·18-s + (6.59 − 3.80i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (0.918 − 0.530i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.155 − 0.0896i)11-s − 0.288·12-s + (0.951 − 0.307i)13-s + 0.749·14-s + (−0.125 + 0.216i)16-s + (−0.697 − 1.20i)17-s − 0.235i·18-s + (1.51 − 0.873i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.947 - 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.620207712\)
\(L(\frac12)\) \(\approx\) \(2.620207712\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.43 + 1.10i)T \)
good7 \( 1 + (-2.42 + 1.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.515 + 0.297i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.87 + 4.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.59 + 3.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.32 - 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 + (-8.98 - 5.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.98 + 2.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.89iT - 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + (8.40 - 4.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.41 + 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.80 - 3.93i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.11 - 0.642i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.5iT - 73T^{2} \)
79 \( 1 - 1.83T + 79T^{2} \)
83 \( 1 - 4.19iT - 83T^{2} \)
89 \( 1 + (-5.24 - 3.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.6 - 8.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.240950665223724633677192341318, −8.290710354066492136038114746560, −7.57555104200419177366494834169, −6.86434250549553366705564329465, −5.80925237181127970481561734298, −5.16209141267444260277024912773, −4.43881664585588535630497799334, −3.61285302622975875708256716749, −2.54706469932172769303645308424, −0.932841602075847161944805764354, 1.29233410078499573078076585713, 2.02657231364230006086304625013, 3.25075550653484871823735844093, 4.28047664071806896838803657037, 5.12001314045449301956005604827, 5.93795046011935623499869109800, 6.49846122549182997848867245344, 7.61772901215829808831823477133, 8.314609688435680999512780969392, 9.018324510401846746900017196204

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