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Properties

Label 2-825-15.8-c1-0-28
Degree $2$
Conductor $825$
Sign $-0.920 - 0.391i$
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  + (1.70 + 1.70i)2-s + (1 + 1.41i)3-s + 3.82i·4-s + (−0.707 + 4.12i)6-s + (0.585 − 0.585i)7-s + (−3.12 + 3.12i)8-s + (−1.00 + 2.82i)9-s + i·11-s + (−5.41 + 3.82i)12-s + (2 + 2i)13-s + 2·14-s − 2.99·16-s + (−2.82 − 2.82i)17-s + (−6.53 + 3.12i)18-s − 2.82i·19-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + (0.577 + 0.816i)3-s + 1.91i·4-s + (−0.288 + 1.68i)6-s + (0.221 − 0.221i)7-s + (−1.10 + 1.10i)8-s + (−0.333 + 0.942i)9-s + 0.301i·11-s + (−1.56 + 1.10i)12-s + (0.554 + 0.554i)13-s + 0.534·14-s − 0.749·16-s + (−0.685 − 0.685i)17-s + (−1.54 + 0.735i)18-s − 0.648i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.920 - 0.391i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701702 + 3.44618i\)
\(L(\frac12)\) \(\approx\) \(0.701702 + 3.44618i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 - 1.41i)T \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-1.70 - 1.70i)T + 2iT^{2} \)
7 \( 1 + (-0.585 + 0.585i)T - 7iT^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (2.82 + 2.82i)T + 17iT^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + (-5.24 + 5.24i)T - 23iT^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + (-5.82 + 5.82i)T - 37iT^{2} \)
41 \( 1 + 3.65iT - 41T^{2} \)
43 \( 1 + (-8.24 - 8.24i)T + 43iT^{2} \)
47 \( 1 + (1.24 + 1.24i)T + 47iT^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + (-3.58 + 3.58i)T - 67iT^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 5.17iT - 79T^{2} \)
83 \( 1 + (5.07 - 5.07i)T - 83iT^{2} \)
89 \( 1 - 1.65T + 89T^{2} \)
97 \( 1 + (-0.656 + 0.656i)T - 97iT^{2} \)
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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

−10.83176981362440964680418515759, −9.329066046697365392249111633523, −8.897229547945978134692646390364, −7.73449966890276519888319667550, −7.13787540652739811304912687953, −6.13016205372987148376560684549, −5.07195815877535660133691697046, −4.46278373432925921509165297041, −3.68488520158486158013217938576, −2.49539654128689761425007508387, 1.26074146044966490430711807750, 2.21677995884008600009283489858, 3.31647290635706966181084980811, 3.95687024356980409401271330870, 5.39910480605603231706708573348, 5.98283576420688931687240586682, 7.18968710036931506022565682671, 8.239277637945841508555366673249, 9.097883990172080387564160563169, 10.08653341275712716568461206556

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