Properties

Label 2-825-5.4-c3-0-25
Degree $2$
Conductor $825$
Sign $-0.894 + 0.447i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.90i·2-s + 3i·3-s + 4.36·4-s − 5.71·6-s + 22.9i·7-s + 23.5i·8-s − 9·9-s + 11·11-s + 13.0i·12-s + 66.7i·13-s − 43.6·14-s − 10.0·16-s + 3.45i·17-s − 17.1i·18-s − 78.2·19-s + ⋯
L(s)  = 1  + 0.674i·2-s + 0.577i·3-s + 0.545·4-s − 0.389·6-s + 1.23i·7-s + 1.04i·8-s − 0.333·9-s + 0.301·11-s + 0.315i·12-s + 1.42i·13-s − 0.834·14-s − 0.156·16-s + 0.0493i·17-s − 0.224i·18-s − 0.944·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.022597386\)
\(L(\frac12)\) \(\approx\) \(2.022597386\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 - 1.90iT - 8T^{2} \)
7 \( 1 - 22.9iT - 343T^{2} \)
13 \( 1 - 66.7iT - 2.19e3T^{2} \)
17 \( 1 - 3.45iT - 4.91e3T^{2} \)
19 \( 1 + 78.2T + 6.85e3T^{2} \)
23 \( 1 - 12.2iT - 1.21e4T^{2} \)
29 \( 1 - 31.1T + 2.43e4T^{2} \)
31 \( 1 - 247.T + 2.97e4T^{2} \)
37 \( 1 + 304. iT - 5.06e4T^{2} \)
41 \( 1 + 29.8T + 6.89e4T^{2} \)
43 \( 1 - 269. iT - 7.95e4T^{2} \)
47 \( 1 + 225. iT - 1.03e5T^{2} \)
53 \( 1 + 16.8iT - 1.48e5T^{2} \)
59 \( 1 + 28.0T + 2.05e5T^{2} \)
61 \( 1 + 853.T + 2.26e5T^{2} \)
67 \( 1 - 36.7iT - 3.00e5T^{2} \)
71 \( 1 - 23.8T + 3.57e5T^{2} \)
73 \( 1 - 707. iT - 3.89e5T^{2} \)
79 \( 1 - 412.T + 4.93e5T^{2} \)
83 \( 1 + 552. iT - 5.71e5T^{2} \)
89 \( 1 - 1.49e3T + 7.04e5T^{2} \)
97 \( 1 + 1.19e3iT - 9.12e5T^{2} \)
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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

−10.29410599905593013257097610385, −9.176912140013841068536000832322, −8.737131293096025975303248973267, −7.79143550542417332935864754273, −6.59642025071383759017530636093, −6.14747147319619796991084826886, −5.14683414185642512479679067533, −4.18431131880339789659203921616, −2.74450066485110859435988442785, −1.87469471772198786620980330762, 0.51543036041311502224827911407, 1.35308911784514062930252593028, 2.65367924798567292715516736935, 3.53427035484322211936704981873, 4.63987811616725456123901048353, 6.10369390968730866312389001709, 6.75169491320244976992993318647, 7.62577247413108491147666317700, 8.303266613297277749813617396556, 9.654898789274835125365026796146

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