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Properties

Label 2-3900-13.10-c1-0-10
Degree $2$
Conductor $3900$
Sign $-0.00641 - 0.999i$
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 + (−3.40 − 1.96i)7-s + (−0.499 + 0.866i)9-s + (−1.14 + 0.658i)11-s + (−1.86 − 3.08i)13-s + (0.276 − 0.478i)17-s + (4.69 + 2.71i)19-s − 3.93i·21-s + (0.237 + 0.411i)23-s − 0.999·27-s + (1.53 + 2.66i)29-s − 3.49i·31-s + (−1.14 − 0.658i)33-s + (−0.407 + 0.235i)37-s + (1.74 − 3.15i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−1.28 − 0.743i)7-s + (−0.166 + 0.288i)9-s + (−0.344 + 0.198i)11-s + (−0.516 − 0.856i)13-s + (0.0670 − 0.116i)17-s + (1.07 + 0.622i)19-s − 0.858i·21-s + (0.0495 + 0.0858i)23-s − 0.192·27-s + (0.285 + 0.495i)29-s − 0.628i·31-s + (−0.198 − 0.114i)33-s + (−0.0669 + 0.0386i)37-s + (0.279 − 0.505i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00641 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00641 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.00641 - 0.999i$
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: \((2,\ 3900,\ (\ :1/2),\ -0.00641 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.126003698\)
\(L(\frac12)\) \(\approx\) \(1.126003698\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 3.08i)T \)
good7 \( 1 + (3.40 + 1.96i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.14 - 0.658i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.276 + 0.478i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.69 - 2.71i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.237 - 0.411i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.53 - 2.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.49iT - 31T^{2} \)
37 \( 1 + (0.407 - 0.235i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.53 + 1.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.697 - 1.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.71iT - 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
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 \( 1 + (11.0 - 6.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.51 + 2.60i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.95iT - 73T^{2} \)
79 \( 1 - 1.19T + 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + (15.5 - 8.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.54 - 2.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.777059770140523112408046165537, −7.67937619539275602996391081830, −7.44980061787250512831258343408, −6.43000971073406276309042680245, −5.65427364241300805953552412116, −4.87773098319819880186187243483, −3.90757830391030393906868177708, −3.24252718899186789632321650842, −2.56397831491552839003405221624, −0.953003913406273693413203856270, 0.36953611829227979198718439412, 1.86059935427475747403405574777, 2.81128772198957726769084553441, 3.32234894893631824508133535938, 4.49558478426547441870686959947, 5.46119981669050370152160332966, 6.13714367780032557025648087465, 6.91794472301122201808899530864, 7.35991420396936599085655731341, 8.441379960131658240533680098892

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