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Properties

Label 2-3900-13.10-c1-0-19
Degree $2$
Conductor $3900$
Sign $0.997 + 0.0751i$
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 + (−2.65 − 1.53i)7-s + (−0.499 + 0.866i)9-s + (1.31 − 0.758i)11-s + (−3.22 + 1.60i)13-s + (−0.193 + 0.335i)17-s + (0.595 + 0.343i)19-s − 3.06i·21-s + (−1.64 − 2.85i)23-s − 0.999·27-s + (4.84 + 8.39i)29-s − 5.19i·31-s + (1.31 + 0.758i)33-s + (4.76 − 2.75i)37-s + (−3.00 − 1.99i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−1.00 − 0.579i)7-s + (−0.166 + 0.288i)9-s + (0.396 − 0.228i)11-s + (−0.895 + 0.445i)13-s + (−0.0470 + 0.0814i)17-s + (0.136 + 0.0788i)19-s − 0.669i·21-s + (−0.343 − 0.595i)23-s − 0.192·27-s + (0.900 + 1.55i)29-s − 0.932i·31-s + (0.228 + 0.132i)33-s + (0.783 − 0.452i)37-s + (−0.481 − 0.319i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.997 + 0.0751i$
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: \((2,\ 3900,\ (\ :1/2),\ 0.997 + 0.0751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.581473027\)
\(L(\frac12)\) \(\approx\) \(1.581473027\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.22 - 1.60i)T \)
good7 \( 1 + (2.65 + 1.53i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.31 + 0.758i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.193 - 0.335i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.595 - 0.343i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.64 + 2.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.84 - 8.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + (-4.76 + 2.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.27 + 1.89i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.69 - 2.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.47iT - 47T^{2} \)
53 \( 1 - 0.937T + 53T^{2} \)
59 \( 1 + (-3.14 - 1.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.05 + 5.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.1 + 7.60i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.2 - 5.89i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.698iT - 73T^{2} \)
79 \( 1 + 4.96T + 79T^{2} \)
83 \( 1 + 2.56iT - 83T^{2} \)
89 \( 1 + (-8.80 + 5.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 6.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.545787166049891831387772914232, −7.74909516133191830723212566629, −6.89216719718597695249549337177, −6.42617581695828757137765035442, −5.41476704788144271999373724550, −4.53176899155795981412113880133, −3.84043094690216038192394296988, −3.07703306798439193327957245086, −2.14760482156658383413397377793, −0.61655419510718887611013863000, 0.78565559274312532724037502964, 2.19845302988935808712899655944, 2.83287701501737502656286343698, 3.70505472588850745940153772947, 4.73238198961860470650863401841, 5.65625139990096815408062326619, 6.36256779268648437288126472831, 6.96193296414297679901682795524, 7.77338143301783281240201893453, 8.404487611451155392705211688051

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