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Properties

Label 2-3900-13.4-c1-0-28
Degree $2$
Conductor $3900$
Sign $0.252 + 0.967i$
Analytic cond. $31.1416$
Root an. cond. $5.58047$
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)3-s + (−3.12 + 1.80i)7-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s + (1 − 3.46i)13-s + (−1 − 1.73i)17-s + (−0.122 + 0.0707i)19-s + 3.60i·21-s + (3.62 − 6.27i)23-s − 0.999·27-s + (−2.62 + 4.54i)29-s + 3.46i·31-s + (3 − 1.73i)33-s + (6.24 + 3.60i)37-s + (−2.49 − 2.59i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−1.18 + 0.681i)7-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s + (0.277 − 0.960i)13-s + (−0.242 − 0.420i)17-s + (−0.0281 + 0.0162i)19-s + 0.786i·21-s + (0.755 − 1.30i)23-s − 0.192·27-s + (−0.486 + 0.843i)29-s + 0.622i·31-s + (0.522 − 0.301i)33-s + (1.02 + 0.592i)37-s + (−0.400 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.612686784\)
\(L(\frac12)\) \(\approx\) \(1.612686784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-1 + 3.46i)T \)
good7 \( 1 + (3.12 - 1.80i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.122 - 0.0707i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.62 + 6.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.62 - 4.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (-6.24 - 3.60i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.24 + 3.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.12 - 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.74iT - 47T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 + (-1.62 + 0.936i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.12 + 3.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.24 + 3.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.4iT - 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 + 5.05iT - 83T^{2} \)
89 \( 1 + (-10.6 - 6.13i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.2 + 7.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.433734364401354210739118700102, −7.52730086195527171909693098445, −6.65852694110084283871342604093, −6.40130893312840737569475171468, −5.43337145651644936142373180649, −4.52985107605234308808467850620, −3.34632605266378184913671718659, −2.91435332397011508110545355556, −1.81476195397015990036862464011, −0.52765537550494954833769815309, 1.01148987536702612976291608543, 2.29514807043844945756617553056, 3.56099765250542611599915827905, 3.74643817594934564790417892542, 4.64889425366845004493025217479, 5.83950456787539991927882425177, 6.39280564143637140176245395066, 7.10251437718662593733972826308, 7.88600548813077630359893290179, 8.883236350985597995147768407943

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