Properties

Label 2-825-55.49-c1-0-0
Degree $2$
Conductor $825$
Sign $-0.480 - 0.877i$
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  + (−1.53 + 0.5i)2-s + (−0.587 + 0.809i)3-s + (0.5 − 0.363i)4-s + (0.5 − 1.53i)6-s + (−3.07 − 4.23i)7-s + (1.31 − 1.80i)8-s + (−0.309 − 0.951i)9-s + (−1.23 − 3.07i)11-s + 0.618i·12-s + (−3.07 + i)13-s + (6.85 + 4.97i)14-s + (−1.50 + 4.61i)16-s + (−1.90 − 0.618i)17-s + (0.951 + 1.30i)18-s + (4.04 + 2.93i)19-s + ⋯
L(s)  = 1  + (−1.08 + 0.353i)2-s + (−0.339 + 0.467i)3-s + (0.250 − 0.181i)4-s + (0.204 − 0.628i)6-s + (−1.16 − 1.60i)7-s + (0.464 − 0.639i)8-s + (−0.103 − 0.317i)9-s + (−0.372 − 0.927i)11-s + 0.178i·12-s + (−0.853 + 0.277i)13-s + (1.83 + 1.33i)14-s + (−0.375 + 1.15i)16-s + (−0.461 − 0.149i)17-s + (0.224 + 0.308i)18-s + (0.928 + 0.674i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.480 - 0.877i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.480 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130329 + 0.219913i\)
\(L(\frac12)\) \(\approx\) \(0.130329 + 0.219913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
11 \( 1 + (1.23 + 3.07i)T \)
good2 \( 1 + (1.53 - 0.5i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (3.07 + 4.23i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (3.07 - i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.90 + 0.618i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.04 - 2.93i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.61iT - 23T^{2} \)
29 \( 1 + (-0.690 + 0.502i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.16 - 6.65i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.865 - 1.19i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.28 + 6.74i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.09iT - 43T^{2} \)
47 \( 1 + (-3.57 + 4.92i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.11 + 1.66i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.42 + 3.94i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.16 + 6.65i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + (2.47 - 7.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.66 - 3.66i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.954 + 2.93i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-7.33 - 2.38i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.14T + 89T^{2} \)
97 \( 1 + (-2.40 + 0.781i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.07683137779315326337391971432, −9.893657640689639916558964862505, −8.951136542964819010879190927812, −7.937682394137391711020511832762, −7.11682743979635888002901651495, −6.57444513891492288650005579446, −5.25673629902215736935966287159, −4.01884336748027893554911683521, −3.28392685147656144055755253851, −0.941141408091541424160845685062, 0.24614996010214784158407187670, 2.15479521788513146443288003367, 2.75221317055096816481366266133, 4.79213554197640517944052028763, 5.57996285639355172635535443986, 6.63626639729345325267734811613, 7.50112554543792140358963508730, 8.477133599802868418293907817247, 9.236850424574460986344366909405, 9.814144411892151382750703785697

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