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Properties

Label 2-825-15.2-c1-0-30
Degree $2$
Conductor $825$
Sign $0.662 - 0.749i$
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  + (0.292 − 0.292i)2-s + (1.70 + 0.292i)3-s + 1.82i·4-s + (0.585 − 0.414i)6-s + (0.585 + 0.585i)7-s + (1.12 + 1.12i)8-s + (2.82 + i)9-s + i·11-s + (−0.535 + 3.12i)12-s + (3.41 − 3.41i)13-s + 0.343·14-s − 3·16-s + (−2 + 2i)17-s + (1.12 − 0.535i)18-s + 0.828i·19-s + ⋯
L(s)  = 1  + (0.207 − 0.207i)2-s + (0.985 + 0.169i)3-s + 0.914i·4-s + (0.239 − 0.169i)6-s + (0.221 + 0.221i)7-s + (0.396 + 0.396i)8-s + (0.942 + 0.333i)9-s + 0.301i·11-s + (−0.154 + 0.901i)12-s + (0.946 − 0.946i)13-s + 0.0917·14-s − 0.750·16-s + (−0.485 + 0.485i)17-s + (0.264 − 0.126i)18-s + 0.190i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33252 + 1.05187i\)
\(L(\frac12)\) \(\approx\) \(2.33252 + 1.05187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.292i)T \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-0.292 + 0.292i)T - 2iT^{2} \)
7 \( 1 + (-0.585 - 0.585i)T + 7iT^{2} \)
13 \( 1 + (-3.41 + 3.41i)T - 13iT^{2} \)
17 \( 1 + (2 - 2i)T - 17iT^{2} \)
19 \( 1 - 0.828iT - 19T^{2} \)
23 \( 1 + (0.828 + 0.828i)T + 23iT^{2} \)
29 \( 1 + 8.82T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.65 - 5.65i)T + 37iT^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + (-4.58 + 4.58i)T - 43iT^{2} \)
47 \( 1 + (4.82 - 4.82i)T - 47iT^{2} \)
53 \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \)
59 \( 1 + 2.34T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (2.58 + 2.58i)T + 67iT^{2} \)
71 \( 1 + 9.65iT - 71T^{2} \)
73 \( 1 + (-4.58 + 4.58i)T - 73iT^{2} \)
79 \( 1 + 4.82iT - 79T^{2} \)
83 \( 1 + (4.24 + 4.24i)T + 83iT^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (1.65 + 1.65i)T + 97iT^{2} \)
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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

−10.39508572491186295960126115491, −9.290224564161907886059759355472, −8.545613092414977970698566268266, −7.958278690197712304800656593081, −7.21844181507616640288459311071, −5.93938115262878988423519633726, −4.60568035347145581432656853430, −3.79497754344575628424864325554, −2.94137717985763612482867840600, −1.85448241237527522198487910563, 1.21422719968621377772516338099, 2.33318368269958098806570792465, 3.81324927523008027893747198673, 4.56335765375465262735427883928, 5.82523521115609728910399684985, 6.68615218990765444464701753032, 7.46503078279020713921396661532, 8.518217032122869448609236330899, 9.288819367208929568084310883013, 9.861296775093620296521680255026

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