Properties

Label 2-1950-65.64-c1-0-31
Degree $2$
Conductor $1950$
Sign $0.992 + 0.124i$
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  + 2-s + i·3-s + 4-s + i·6-s + 3·7-s + 8-s − 9-s − 5i·11-s + i·12-s + (2 − 3i)13-s + 3·14-s + 16-s + 7i·17-s − 18-s − 4i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.13·7-s + 0.353·8-s − 0.333·9-s − 1.50i·11-s + 0.288i·12-s + (0.554 − 0.832i)13-s + 0.801·14-s + 0.250·16-s + 1.69i·17-s − 0.235·18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.204240674\)
\(L(\frac12)\) \(\approx\) \(3.204240674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + (-2 + 3i)T \)
good7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 11iT - 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.900708294718218593908172421737, −8.362753038381910379653896399934, −7.84962674250234991788385617270, −6.41317365972876743535271740190, −5.91875119374335620317214570393, −5.05038657882760264012725781375, −4.27924155719094403777257246368, −3.44662618665729491112720039250, −2.49153215459591138965778677445, −1.02595622375540356416283476511, 1.45491289552526550486262302402, 2.09336847200769477098002337447, 3.35178702918451943111584424923, 4.53750356151393784038598312343, 4.98319389822163979942693905356, 5.95174744379117662786101475525, 7.05557250673826687196833987821, 7.33189067866850681642682591467, 8.221386122725656731195566504438, 9.174083056813021041848900294074

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