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Properties

Label 2-825-275.196-c1-0-37
Degree $2$
Conductor $825$
Sign $-0.911 - 0.411i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 1.31i)2-s + 3-s + (0.927 + 2.85i)4-s + (−0.690 + 2.12i)5-s + (−1.80 − 1.31i)6-s + (−0.309 − 0.224i)7-s + (0.690 − 2.12i)8-s + 9-s + (4.04 − 2.93i)10-s + (−0.309 + 3.30i)11-s + (0.927 + 2.85i)12-s − 6·13-s + (0.263 + 0.812i)14-s + (−0.690 + 2.12i)15-s + (0.809 − 0.587i)16-s + (−1.42 − 1.03i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.929i)2-s + 0.577·3-s + (0.463 + 1.42i)4-s + (−0.309 + 0.951i)5-s + (−0.738 − 0.536i)6-s + (−0.116 − 0.0848i)7-s + (0.244 − 0.751i)8-s + 0.333·9-s + (1.27 − 0.929i)10-s + (−0.0931 + 0.995i)11-s + (0.267 + 0.823i)12-s − 1.66·13-s + (0.0705 + 0.217i)14-s + (−0.178 + 0.549i)15-s + (0.202 − 0.146i)16-s + (−0.346 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.911 - 0.411i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (0.690 - 2.12i)T \)
11 \( 1 + (0.309 - 3.30i)T \)
good2 \( 1 + (1.80 + 1.31i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.309 + 0.224i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1.42 + 1.03i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.35 + 4.16i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.07 - 3.30i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.26 + 3.88i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.454 + 1.40i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.42 + 7.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (5.04 + 3.66i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 + 3.09T + 47T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.527 - 1.62i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + (0.690 - 0.502i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.454 - 1.40i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1 - 0.726i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4 - 2.90i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.35 + 6.79i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (5.11 - 15.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (9.28 - 6.74i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.863266983934447757056653154932, −9.231149950003862706587472396063, −8.119038016520649466976253233461, −7.34798061559415692634722596141, −6.94514162048783402392379798476, −5.09463934852671097936617451596, −3.77854048218075653157474898494, −2.63208539138304102959892375959, −2.09901032974110689256623466548, 0, 1.54079653817605911296545185860, 3.22242610827965033039974850356, 4.63652699871767078287635805360, 5.64390027508103111346320025122, 6.71296247319353262223706175768, 7.58763221538314610853346521415, 8.379056541163967190606843350910, 8.637784750206458650933292532474, 9.767427407759499644421201614328

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