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Properties

Label 2-1950-13.3-c1-0-31
Degree $2$
Conductor $1950$
Sign $-0.597 + 0.802i$
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + (1.78 − 3.08i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.280 + 0.486i)11-s + 0.999·12-s + (2.84 − 2.21i)13-s − 3.56·14-s + (−0.5 − 0.866i)16-s + (1.56 − 2.70i)17-s + 0.999·18-s + (1.21 − 2.11i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s + (0.673 − 1.16i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.0846 + 0.146i)11-s + 0.288·12-s + (0.788 − 0.615i)13-s − 0.951·14-s + (−0.125 − 0.216i)16-s + (0.378 − 0.655i)17-s + 0.235·18-s + (0.279 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.597 + 0.802i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.447478489\)
\(L(\frac12)\) \(\approx\) \(1.447478489\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-2.84 + 2.21i)T \)
good7 \( 1 + (-1.78 + 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.280 - 0.486i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.56 + 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.21 + 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.84 - 6.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.561 + 0.972i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (0.280 + 0.486i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.56 - 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.219 - 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + (-5.12 + 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.842 + 1.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.90 + 10.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.28 - 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 6.56T + 83T^{2} \)
89 \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.40 + 2.43i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.938108550006578133921567583854, −8.006144618475602204073708783732, −7.49060626564070976323470121094, −6.80862558046174272395183774030, −5.60970730723455523042275500467, −4.78765305051872420537478003421, −3.80294833043411453821392761823, −2.87049496051891924484942461643, −1.47271067482213714227876390537, −0.75723357489625124904657591091, 1.24328915854516000488841986346, 2.54285196020158346336533423871, 3.88831806205572272985996982160, 4.76214168797417561251119486865, 5.62624651169231994252144084049, 6.13366966288620055430666944235, 7.04305008290001033120398817879, 8.162182174250796946559074754440, 8.702512479662282610747545823674, 9.151615281861327952083884363754

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