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Properties

Label 2-1950-65.49-c1-0-15
Degree $2$
Conductor $1950$
Sign $0.896 + 0.442i$
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 + (1.79 − 3.10i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−4.72 + 2.72i)11-s − 0.999i·12-s + (−3.60 + 0.0664i)13-s − 3.59·14-s + (−0.5 − 0.866i)16-s + (6.33 + 3.65i)17-s − 0.999·18-s + (3.34 + 1.93i)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 + (0.678 − 1.17i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−1.42 + 0.821i)11-s − 0.288i·12-s + (−0.999 + 0.0184i)13-s − 0.959·14-s + (−0.125 − 0.216i)16-s + (1.53 + 0.886i)17-s − 0.235·18-s + (0.767 + 0.443i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.896 + 0.442i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.896 + 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.083450257\)
\(L(\frac12)\) \(\approx\) \(1.083450257\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 0.0664i)T \)
good7 \( 1 + (-1.79 + 3.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.72 - 2.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-6.33 - 3.65i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.34 - 1.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 0.929i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.67 + 4.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.23iT - 31T^{2} \)
37 \( 1 + (-1.06 - 1.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.33 - 3.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.10 - 4.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.07T + 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + (-0.237 - 0.137i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.06 + 3.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.39 + 12.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.3 - 7.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 - 7.15T + 83T^{2} \)
89 \( 1 + (8.30 - 4.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.954 - 1.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.722743393711358640183491344014, −7.982795844082745981286585125765, −7.83993010985233030243247221071, −7.07372829087360110738657276643, −5.69118599306534063640009686547, −4.93638741740929237238844887933, −4.24171927028244991210094475494, −3.22396626174201977374074310775, −1.99891547896166752714384581945, −0.789840841739644763751363120838, 0.72270717882046090222917853684, 2.23572019295249858676719182955, 3.16863184801323774913650257341, 5.04232550672493143328492010837, 5.28892211260060106852303651161, 5.78969756650468706023132804240, 7.18511624098734883564080909072, 7.54700264133915758822766963294, 8.360475536926023526768165792358, 9.101223260622968859831747391399

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