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Properties

Label 2-1950-65.49-c1-0-40
Degree $2$
Conductor $1950$
Sign $-0.545 + 0.838i$
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 + (1.46 − 2.53i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−0.992 + 0.573i)11-s − 0.999i·12-s + (−3.08 + 1.86i)13-s + 2.93·14-s + (−0.5 − 0.866i)16-s + (−0.478 − 0.276i)17-s + 0.999·18-s + (−0.723 − 0.417i)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 + (0.554 − 0.959i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.299 + 0.172i)11-s − 0.288i·12-s + (−0.856 + 0.516i)13-s + 0.783·14-s + (−0.125 − 0.216i)16-s + (−0.116 − 0.0670i)17-s + 0.235·18-s + (−0.165 − 0.0958i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.545 + 0.838i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05258983158\)
\(L(\frac12)\) \(\approx\) \(0.05258983158\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.08 - 1.86i)T \)
good7 \( 1 + (-1.46 + 2.53i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.992 - 0.573i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.478 + 0.276i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.723 + 0.417i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.859 - 0.496i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.03 + 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.36iT - 31T^{2} \)
37 \( 1 + (4.04 + 7.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.67 - 5.58i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.82 + 3.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 3.56iT - 53T^{2} \)
59 \( 1 + (-4.75 - 2.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.43 - 2.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.13 + 8.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.85 - 5.69i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.19T + 73T^{2} \)
79 \( 1 + 1.68T + 79T^{2} \)
83 \( 1 + 8.77T + 83T^{2} \)
89 \( 1 + (-3.93 + 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.30 + 9.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.800183724246449445673621260949, −7.971594711818260620355694775930, −7.14461596369873267017592011003, −6.71737441647953369360323963827, −5.57870657676565069595986816151, −4.82567133456577052130900147729, −4.29120515380176685455585223317, −3.25511793243621752905550488228, −1.75376432649991535748656086351, −0.01701668494604432881328023677, 1.63735962728241484229058542178, 2.48865381665560229435569301823, 3.51800141134890447993372972252, 4.90609914455299269080892514548, 5.17447801005259581638274986582, 6.10571076360723253924704111512, 6.96797107269236748779419376944, 8.074323741253851057449965977166, 8.574856875422352436019344829138, 9.686944547646811322335883724064

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