↑

Properties

Label 2-3675-735.149-c0-0-0
Degree $2$
Conductor $3675$
Sign $0.121 - 0.992i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.930 + 0.365i)3-s + (−0.826 − 0.563i)4-s + (0.680 + 0.733i)7-s + (0.733 + 0.680i)9-s + (−0.563 − 0.826i)12-s + (−1.92 + 0.440i)13-s + (0.365 + 0.930i)16-s + (−0.733 + 1.26i)19-s + (0.365 + 0.930i)21-s + (0.433 + 0.900i)27-s + (−0.149 − 0.988i)28-s + (0.900 + 1.56i)31-s + (−0.222 − 0.974i)36-s + (−0.0841 − 0.123i)37-s + (−1.95 − 0.294i)39-s + ⋯
L(s)  = 1  + (0.930 + 0.365i)3-s + (−0.826 − 0.563i)4-s + (0.680 + 0.733i)7-s + (0.733 + 0.680i)9-s + (−0.563 − 0.826i)12-s + (−1.92 + 0.440i)13-s + (0.365 + 0.930i)16-s + (−0.733 + 1.26i)19-s + (0.365 + 0.930i)21-s + (0.433 + 0.900i)27-s + (−0.149 − 0.988i)28-s + (0.900 + 1.56i)31-s + (−0.222 − 0.974i)36-s + (−0.0841 − 0.123i)37-s + (−1.95 − 0.294i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.121 - 0.992i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.121 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292964979\)
\(L(\frac12)\) \(\approx\) \(1.292964979\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.930 - 0.365i)T \)
5 \( 1 \)
7 \( 1 + (-0.680 - 0.733i)T \)
good2 \( 1 + (0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.0747 + 0.997i)T^{2} \)
13 \( 1 + (1.92 - 0.440i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.733 - 1.26i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.0841 + 0.123i)T + (-0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.347 + 0.277i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.826 + 0.563i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (-1.36 + 0.930i)T + (0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.632 - 0.365i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.0440 - 0.142i)T + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 + 1.91iT - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.835541266051508124932872955714, −8.340111241998856037016227340794, −7.67065746530546951679786264428, −6.73527549272771805545160168531, −5.64889899770991479151977867827, −4.85103871115749191058727532407, −4.53396501022864493452325428751, −3.46859766103152780911862404421, −2.35783217461158941012352515521, −1.65661580641095062242087669628, 0.66832242621897235069686670292, 2.25013184988030311807809672079, 2.88188671718926911924893254470, 4.02098810494936283336835934226, 4.54268931541045729443492061679, 5.22545298800152766300720995277, 6.67071104041826629586576745958, 7.32852010353103292183076515254, 7.87056967722081969444272136606, 8.369069608930672809573090734297

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