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Properties

Label 2-825-33.32-c1-0-50
Degree $2$
Conductor $825$
Sign $-0.486 + 0.873i$
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  + 1.37·2-s + (−1.70 − 0.329i)3-s − 0.122·4-s + (−2.32 − 0.452i)6-s + 3.12i·7-s − 2.90·8-s + (2.78 + 1.12i)9-s + (1.03 − 3.15i)11-s + (0.207 + 0.0403i)12-s − 5.18i·13-s + 4.27i·14-s − 3.74·16-s − 4.02·17-s + (3.81 + 1.53i)18-s − 4.65i·19-s + ⋯
L(s)  = 1  + 0.968·2-s + (−0.981 − 0.190i)3-s − 0.0611·4-s + (−0.951 − 0.184i)6-s + 1.18i·7-s − 1.02·8-s + (0.927 + 0.374i)9-s + (0.310 − 0.950i)11-s + (0.0599 + 0.0116i)12-s − 1.43i·13-s + 1.14i·14-s − 0.935·16-s − 0.976·17-s + (0.898 + 0.362i)18-s − 1.06i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.486 + 0.873i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.425020 - 0.722723i\)
\(L(\frac12)\) \(\approx\) \(0.425020 - 0.722723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.329i)T \)
5 \( 1 \)
11 \( 1 + (-1.03 + 3.15i)T \)
good2 \( 1 - 1.37T + 2T^{2} \)
7 \( 1 - 3.12iT - 7T^{2} \)
13 \( 1 + 5.18iT - 13T^{2} \)
17 \( 1 + 4.02T + 17T^{2} \)
19 \( 1 + 4.65iT - 19T^{2} \)
23 \( 1 + 2.16iT - 23T^{2} \)
29 \( 1 + 0.904T + 29T^{2} \)
31 \( 1 + 2.74T + 31T^{2} \)
37 \( 1 - 4.06T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 5.43iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + 1.81iT - 53T^{2} \)
59 \( 1 + 7.97iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 + 6.65T + 67T^{2} \)
71 \( 1 - 2.06iT - 71T^{2} \)
73 \( 1 - 8.85iT - 73T^{2} \)
79 \( 1 - 0.821iT - 79T^{2} \)
83 \( 1 - 4.95T + 83T^{2} \)
89 \( 1 + 2.39iT - 89T^{2} \)
97 \( 1 - 7.06T + 97T^{2} \)
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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

−10.13378178746356467832908606097, −8.948991816134766401153300664207, −8.425168985534498941444964628299, −6.94478164376904187983844676825, −6.08752063862960934131555091117, −5.45510931321140387485859222578, −4.86164886869788620453631077830, −3.59649115277956784508645266427, −2.46340726172978254304833451039, −0.34143102148943200270211721639, 1.64710550697622761670271289482, 3.68262709867824321297206652169, 4.35897000479978029151172725475, 4.84622818338554697546148053369, 6.14111626732678573752885079648, 6.71674355481717781169681294419, 7.58801642985254048269322154424, 9.130040965687259651828101150535, 9.734480955299701256215279576060, 10.65220764387025167040304534954

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