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Properties

Label 2-3900-13.10-c1-0-11
Degree $2$
Conductor $3900$
Sign $-0.0602 - 0.998i$
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 + (3.94 + 2.27i)7-s + (−0.499 + 0.866i)9-s + (−3.76 + 2.17i)11-s + (−2.02 − 2.98i)13-s + (−1 + 1.73i)17-s + (2.53 + 1.46i)19-s − 4.55i·21-s + (−1.34 − 2.33i)23-s + 0.999·27-s + (1.72 + 2.98i)29-s − 1.62i·31-s + (3.76 + 2.17i)33-s + (3.76 − 2.17i)37-s + (−1.56 + 3.24i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (1.48 + 0.860i)7-s + (−0.166 + 0.288i)9-s + (−1.13 + 0.654i)11-s + (−0.561 − 0.827i)13-s + (−0.242 + 0.420i)17-s + (0.581 + 0.336i)19-s − 0.993i·21-s + (−0.280 − 0.485i)23-s + 0.192·27-s + (0.320 + 0.554i)29-s − 0.291i·31-s + (0.654 + 0.378i)33-s + (0.618 − 0.356i)37-s + (−0.251 + 0.519i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0602 - 0.998i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ -0.0602 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219876163\)
\(L(\frac12)\) \(\approx\) \(1.219876163\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 + 2.98i)T \)
good7 \( 1 + (-3.94 - 2.27i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.76 - 2.17i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.53 - 1.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.34 + 2.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.72 - 2.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.62iT - 31T^{2} \)
37 \( 1 + (-3.76 + 2.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.44 - 1.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.25 - 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 - 1.44T + 53T^{2} \)
59 \( 1 + (8.15 + 4.71i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.71 + 6.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 + 0.870i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.68 + 0.974i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 9.07T + 79T^{2} \)
83 \( 1 - 17.2iT - 83T^{2} \)
89 \( 1 + (0.445 - 0.257i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.16 - 1.25i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.281627272956584179279758166036, −7.987252624396021488685952386653, −7.44934044109434877968535325425, −6.39271961386608328065964321627, −5.49184915891645379251316572449, −5.10433659040996759648573479901, −4.37432131456982771434599794073, −2.85195225290265729836688168108, −2.23356941260104009803109644383, −1.26947324953158826200750997918, 0.36977913907121264817995874947, 1.65182823593452839221580048696, 2.72534196772433018471203476138, 3.82003146610184726037092439468, 4.66733157931769639218572170586, 5.06640381922548739049736500946, 5.86110432275373120873001659770, 7.03234635134671753528418136374, 7.51499197882569214620190086303, 8.281166757225108260699818511945

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