↑

Properties

Label 2-825-165.47-c1-0-7
Degree $2$
Conductor $825$
Sign $0.0194 + 0.999i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.221 + 1.39i)2-s + (−0.176 + 1.72i)3-s + (−0.00375 − 0.00121i)4-s + (−2.36 − 0.628i)6-s + (−2.59 + 1.32i)7-s + (−1.28 + 2.51i)8-s + (−2.93 − 0.608i)9-s + (−3.16 − 1.00i)11-s + (0.00276 − 0.00625i)12-s + (0.467 − 2.94i)13-s + (−1.27 − 3.91i)14-s + (−3.24 − 2.35i)16-s + (−2.34 + 0.370i)17-s + (1.50 − 3.97i)18-s + (5.08 − 1.65i)19-s + ⋯
L(s)  = 1  + (−0.156 + 0.988i)2-s + (−0.101 + 0.994i)3-s + (−0.00187 − 0.000609i)4-s + (−0.967 − 0.256i)6-s + (−0.980 + 0.499i)7-s + (−0.453 + 0.890i)8-s + (−0.979 − 0.202i)9-s + (−0.953 − 0.302i)11-s + (0.000797 − 0.00180i)12-s + (0.129 − 0.818i)13-s + (−0.340 − 1.04i)14-s + (−0.810 − 0.588i)16-s + (−0.567 + 0.0899i)17-s + (0.353 − 0.936i)18-s + (1.16 − 0.379i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0194 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0194 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.0194 + 0.999i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.0194 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340820 - 0.334244i\)
\(L(\frac12)\) \(\approx\) \(0.340820 - 0.334244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.176 - 1.72i)T \)
5 \( 1 \)
11 \( 1 + (3.16 + 1.00i)T \)
good2 \( 1 + (0.221 - 1.39i)T + (-1.90 - 0.618i)T^{2} \)
7 \( 1 + (2.59 - 1.32i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-0.467 + 2.94i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (2.34 - 0.370i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-5.08 + 1.65i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.65 - 4.65i)T + 23iT^{2} \)
29 \( 1 + (-1.28 + 3.94i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.39 - 4.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (8.36 - 4.25i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.68 - 0.548i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.84 + 3.84i)T - 43iT^{2} \)
47 \( 1 + (3.75 + 1.91i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (8.61 + 1.36i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (2.31 - 7.12i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.294 + 0.213i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.32 - 3.32i)T + 67iT^{2} \)
71 \( 1 + (0.574 - 0.790i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.36 - 8.57i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (-10.2 - 14.1i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.04 + 12.9i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 - 3.56T + 89T^{2} \)
97 \( 1 + (11.7 + 1.86i)T + (92.2 + 29.9i)T^{2} \)
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

−10.78482931620914144707278727720, −9.844341209819969490041026403653, −9.073699216611941931624720634040, −8.378921863415493624114100532653, −7.40727619528803352082450019130, −6.47715616484493474889214448334, −5.46072528864600012660389884713, −5.19357177461587432851269729735, −3.38251806558448425504291822267, −2.78359826119514361275138060271, 0.23191110457459876163533319141, 1.65792320562401209710151191980, 2.72326076125396114430666873499, 3.58008230340969200385681673517, 5.12302164372496693557335657680, 6.39183124501514111369309933459, 6.91489238365849065320006638388, 7.72741093226707315809993866629, 9.037799569340875377518657123518, 9.642507133343949979761809621712

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