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Properties

Label 2-1950-65.4-c1-0-7
Degree $2$
Conductor $1950$
Sign $0.0663 - 0.997i$
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 + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (0.241 + 0.417i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−2.00 − 1.15i)11-s − 0.999i·12-s + (−1.86 + 3.08i)13-s + 0.482·14-s + (−0.5 + 0.866i)16-s + (−5.88 + 3.39i)17-s + 0.999·18-s + (−4.39 + 2.53i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (0.0911 + 0.157i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.605 − 0.349i)11-s − 0.288i·12-s + (−0.516 + 0.856i)13-s + 0.128·14-s + (−0.125 + 0.216i)16-s + (−1.42 + 0.824i)17-s + 0.235·18-s + (−1.00 + 0.582i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.0663 - 0.997i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.0663 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266659741\)
\(L(\frac12)\) \(\approx\) \(1.266659741\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.86 - 3.08i)T \)
good7 \( 1 + (-0.241 - 0.417i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.00 + 1.15i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.88 - 3.39i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.39 - 2.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.98 - 3.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.20 - 7.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.952iT - 31T^{2} \)
37 \( 1 + (-3.09 + 5.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.55 - 4.36i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.78 - 5.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 + 5.81iT - 53T^{2} \)
59 \( 1 + (10.2 - 5.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.68 + 8.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.69 - 6.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.01 + 1.74i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.53T + 73T^{2} \)
79 \( 1 + 9.03T + 79T^{2} \)
83 \( 1 - 7.94T + 83T^{2} \)
89 \( 1 + (7.80 + 4.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.59 - 7.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.226154022662861123278423775460, −8.891823338339977314919814223079, −7.969951246330547846846140895821, −6.98937278473432892210748780129, −6.10349132057427719640094245702, −5.09173873535709673264928547002, −4.36385666270704060377596165920, −3.53571956560385495095430034467, −2.49342639903648603551198180196, −1.69744329265963482506117627508, 0.34874072373444794730460983453, 2.31174384641433197032996891411, 2.91976143334587753784511808447, 4.32399742079816625921325120756, 4.80699740115729633394659178390, 5.87440136167681047600379871091, 6.81011488963401443495226340816, 7.35294711496708980381035273602, 8.091172705053988538087150522395, 8.876048162206647699945791533154

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