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Properties

Label 2-3900-65.4-c1-0-12
Degree $2$
Conductor $3900$
Sign $0.897 - 0.441i$
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  + (−0.866 − 0.5i)3-s + (−1.96 − 3.40i)7-s + (0.499 + 0.866i)9-s + (−1.14 − 0.658i)11-s + (3.08 + 1.86i)13-s + (−0.478 + 0.276i)17-s + (−4.69 + 2.71i)19-s + 3.93i·21-s + (−0.411 − 0.237i)23-s − 0.999i·27-s + (−1.53 + 2.66i)29-s + 3.49i·31-s + (0.658 + 1.14i)33-s + (0.235 − 0.407i)37-s + (−1.74 − 3.15i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (−0.743 − 1.28i)7-s + (0.166 + 0.288i)9-s + (−0.344 − 0.198i)11-s + (0.856 + 0.516i)13-s + (−0.116 + 0.0670i)17-s + (−1.07 + 0.622i)19-s + 0.858i·21-s + (−0.0858 − 0.0495i)23-s − 0.192i·27-s + (−0.285 + 0.495i)29-s + 0.628i·31-s + (0.114 + 0.198i)33-s + (0.0386 − 0.0669i)37-s + (−0.279 − 0.505i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ 0.897 - 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9761300001\)
\(L(\frac12)\) \(\approx\) \(0.9761300001\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-3.08 - 1.86i)T \)
good7 \( 1 + (1.96 + 3.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.14 + 0.658i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.478 - 0.276i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.69 - 2.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.411 + 0.237i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.53 - 2.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.49iT - 31T^{2} \)
37 \( 1 + (-0.235 + 0.407i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.53 - 1.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.20 - 0.697i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.71T + 47T^{2} \)
53 \( 1 - 3.43iT - 53T^{2} \)
59 \( 1 + (11.2 - 6.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.313 - 0.542i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.37 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.51 - 2.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.95T + 73T^{2} \)
79 \( 1 + 1.19T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + (-15.5 - 8.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.62 - 4.54i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.475176630439162432352137890435, −7.65244663596967886341140236479, −6.98532784270855578469078037215, −6.34352817956295816265531903989, −5.79125397070996283097891116629, −4.61787712006465225976082430876, −3.97651946994675311746634602023, −3.18868420540918622278252666480, −1.85929436957316589397134807031, −0.809757677069348835836509121602, 0.40875775379940463074141771289, 2.05375582421393066993658030208, 2.86410574159207953815701885562, 3.81413088442564873105233133479, 4.71511535039560277662117010665, 5.62858916349260941342555256916, 6.05393544932087444343382815796, 6.72368113164976372002591838130, 7.74112995776490021234594940468, 8.594560346447494217187125878533

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