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Properties

Label 2-825-825.479-c1-0-114
Degree $2$
Conductor $825$
Sign $-0.226 - 0.974i$
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  − 2.11i·2-s + (1.16 − 1.27i)3-s − 2.49·4-s + (0.713 − 2.11i)5-s + (−2.70 − 2.47i)6-s + (−4.07 + 2.95i)7-s + 1.04i·8-s + (−0.270 − 2.98i)9-s + (−4.49 − 1.51i)10-s + (−3.05 + 1.29i)11-s + (−2.91 + 3.18i)12-s + (1.02 − 0.743i)13-s + (6.26 + 8.62i)14-s + (−1.87 − 3.38i)15-s − 2.77·16-s + (1.21 − 0.396i)17-s + ⋯
L(s)  = 1  − 1.49i·2-s + (0.674 − 0.738i)3-s − 1.24·4-s + (0.319 − 0.947i)5-s + (−1.10 − 1.01i)6-s + (−1.53 + 1.11i)7-s + 0.368i·8-s + (−0.0900 − 0.995i)9-s + (−1.42 − 0.478i)10-s + (−0.920 + 0.391i)11-s + (−0.840 + 0.919i)12-s + (0.283 − 0.206i)13-s + (1.67 + 2.30i)14-s + (−0.484 − 0.874i)15-s − 0.694·16-s + (0.295 − 0.0961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.226 - 0.974i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.226 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671542 + 0.845385i\)
\(L(\frac12)\) \(\approx\) \(0.671542 + 0.845385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.16 + 1.27i)T \)
5 \( 1 + (-0.713 + 2.11i)T \)
11 \( 1 + (3.05 - 1.29i)T \)
good2 \( 1 + 2.11iT - 2T^{2} \)
7 \( 1 + (4.07 - 2.95i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.02 + 0.743i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.21 + 0.396i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + 3.25iT - 19T^{2} \)
23 \( 1 + (0.243 - 0.750i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + 0.339T + 29T^{2} \)
31 \( 1 + (-7.05 - 5.12i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.14 + 2.95i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.721 + 2.22i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.35T + 43T^{2} \)
47 \( 1 + (3.64 + 11.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.16 + 9.73i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.29 - 3.15i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.00 - 5.50i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.52 + 6.22i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-1.26 - 1.73i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.21 - 3.74i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.82 + 0.916i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.45 + 0.472i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.78 - 0.905i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-15.6 - 5.07i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.786714743871892365291171646719, −8.904391979413638816680946964514, −8.463902578810638972071928353510, −7.03437468757180192712261798118, −6.05205141863603221767854480227, −4.96560036532216545604336729654, −3.50640680286433523474752017364, −2.76284922505352683368658228628, −1.95605542213288537192759177944, −0.45857457363243530730950356233, 2.75468832049898204866824753170, 3.58546711259128245659069647439, 4.64913200336708525654497324283, 6.02373804540749000726968318090, 6.38211591453535111386918359590, 7.54125596476500549642703757804, 7.895291497175262626793271626234, 9.093305411232633792561134206175, 9.975768348471711218026490717661, 10.32062352638423958223131479307

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