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Properties

Label 2-825-33.2-c1-0-64
Degree $2$
Conductor $825$
Sign $-0.212 - 0.977i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.664 − 2.04i)2-s + (−1.72 − 0.200i)3-s + (−2.11 − 1.53i)4-s + (−1.55 + 3.38i)6-s + (1.16 − 1.60i)7-s + (−1.07 + 0.781i)8-s + (2.91 + 0.689i)9-s + (−3.11 + 1.14i)11-s + (3.33 + 3.07i)12-s + (−5.55 − 1.80i)13-s + (−2.50 − 3.44i)14-s + (−0.735 − 2.26i)16-s + (0.00961 + 0.0296i)17-s + (3.34 − 5.50i)18-s + (−3.10 − 4.27i)19-s + ⋯
L(s)  = 1  + (0.469 − 1.44i)2-s + (−0.993 − 0.115i)3-s + (−1.05 − 0.769i)4-s + (−0.633 + 1.38i)6-s + (0.439 − 0.605i)7-s + (−0.380 + 0.276i)8-s + (0.973 + 0.229i)9-s + (−0.938 + 0.345i)11-s + (0.962 + 0.886i)12-s + (−1.54 − 0.500i)13-s + (−0.668 − 0.920i)14-s + (−0.183 − 0.566i)16-s + (0.00233 + 0.00717i)17-s + (0.789 − 1.29i)18-s + (−0.711 − 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.212 - 0.977i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.212 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355213 + 0.440870i\)
\(L(\frac12)\) \(\approx\) \(0.355213 + 0.440870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.200i)T \)
5 \( 1 \)
11 \( 1 + (3.11 - 1.14i)T \)
good2 \( 1 + (-0.664 + 2.04i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.16 + 1.60i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (5.55 + 1.80i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.00961 - 0.0296i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.10 + 4.27i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.23iT - 23T^{2} \)
29 \( 1 + (0.972 + 0.706i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.153 + 0.472i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.98 + 2.16i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.54 + 1.84i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.79iT - 43T^{2} \)
47 \( 1 + (0.700 + 0.963i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-9.74 - 3.16i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.78 + 2.46i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.33 - 2.70i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.68T + 67T^{2} \)
71 \( 1 + (0.561 - 0.182i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.63 + 3.62i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (16.4 + 5.33i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.17 + 9.76i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.78iT - 89T^{2} \)
97 \( 1 + (-5.65 + 17.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.20146629893681659603322718273, −9.278990922389773452213576963841, −7.53015895597625342691657335439, −7.27518112424676408925512390607, −5.62614654020807922045699726375, −4.87498747318379380793732316925, −4.24349376584766633215628427441, −2.82818540900855080418220694965, −1.73734155195494702106207621769, −0.25776016823573922789028674800, 2.25164170001572918643805509166, 4.21787744888056133032339387674, 4.96660188410932552235420636577, 5.55662507253539678864839950319, 6.40883082243734191458104158890, 7.17104143222732851842489482401, 8.016103310630423235238203133721, 8.816591093182781423285087364676, 10.10334801755791835596924001284, 10.71826338396687511346685750187

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