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Properties

Label 2-1950-13.4-c1-0-7
Degree $2$
Conductor $1950$
Sign $0.711 - 0.702i$
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 + (−0.633 + 0.366i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−4.09 − 2.36i)11-s + 0.999·12-s + (−2.59 + 2.5i)13-s + 0.732·14-s + (−0.5 + 0.866i)16-s + (1.13 + 1.96i)17-s + 0.999i·18-s + (−1.09 + 0.633i)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 + (−0.239 + 0.138i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−1.23 − 0.713i)11-s + 0.288·12-s + (−0.720 + 0.693i)13-s + 0.195·14-s + (−0.125 + 0.216i)16-s + (0.275 + 0.476i)17-s + 0.235i·18-s + (−0.251 + 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.711 - 0.702i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7941609029\)
\(L(\frac12)\) \(\approx\) \(0.7941609029\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (-9.06 - 5.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.86 - 5.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.19iT - 47T^{2} \)
53 \( 1 + 0.464T + 53T^{2} \)
59 \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.598 + 1.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.63 - 5.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.09 - 0.633i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.281476123051157153754284441784, −8.364905483913143251447901853029, −7.999473910311689566613891934275, −7.03416989196933358589694681604, −6.34677380575521993574943930300, −5.32376443054217428520377202082, −4.24027703156403770087322553962, −2.95367496971954104060929628859, −2.46784770139233999295301256328, −1.09577675853303949248725458752, 0.37364629654869154881557387663, 2.19564666206182251916709746146, 2.97652263717091875013959214311, 4.24680890881607784523993358335, 5.20613980985200007037442651216, 5.75631243232626555413781224685, 7.05695829845152019930208743135, 7.64643013507142960323129453040, 8.144409329188613570728642150960, 9.361457837363070875867155167040

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