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Properties

Label 2-1950-13.3-c1-0-5
Degree $2$
Conductor $1950$
Sign $0.794 - 0.607i$
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 + (−0.280 + 0.486i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.78 − 3.08i)11-s + 0.999·12-s + (−3.34 + 1.35i)13-s + 0.561·14-s + (−0.5 − 0.866i)16-s + (−2.56 + 4.43i)17-s + 0.999·18-s + (3.28 − 5.68i)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 + (−0.106 + 0.183i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.536 − 0.929i)11-s + 0.288·12-s + (−0.926 + 0.375i)13-s + 0.150·14-s + (−0.125 − 0.216i)16-s + (−0.621 + 1.07i)17-s + 0.235·18-s + (0.752 − 1.30i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.794 - 0.607i$
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: \((2,\ 1950,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5919300747\)
\(L(\frac12)\) \(\approx\) \(0.5919300747\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.34 - 1.35i)T \)
good7 \( 1 + (0.280 - 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.78 + 3.08i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.56 - 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.28 + 5.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.34 + 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.56 - 6.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-1.78 - 3.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.56 + 4.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.28 - 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (3.12 - 5.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.34 - 9.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.40 - 7.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.21 - 7.30i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 - 4.80T + 79T^{2} \)
83 \( 1 - 2.43T + 83T^{2} \)
89 \( 1 + (-3.12 - 5.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.90 - 15.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.182442009464191824573794051655, −8.567956210884958028786907382123, −7.84407836926206298679299303678, −6.92932909537133793011689669447, −6.21810027994020778675612429233, −5.15799545108976493374293148689, −4.37637041838467194253350379986, −3.02943362406411902535666802544, −2.38301992178887271462616337063, −1.03876680964278321437115073621, 0.29038981471890800377337932502, 2.02606642590935384560934506874, 3.28104430876495902267210323984, 4.51936593166882522589388401025, 5.04093014995318019956136220079, 5.91192646066611034894963075966, 6.80728338999127118504455037946, 7.65978957464130509456726830920, 8.062878194232344324433638103416, 9.383546975053277720596759055782

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