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Properties

Label 2-825-15.8-c1-0-18
Degree $2$
Conductor $825$
Sign $0.998 + 0.0618i$
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 + (1.41 + i)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 + (−3.82 + 5.41i)12-s + (2 + 2i)13-s − 2·14-s − 2.99·16-s + (2.82 + 2.82i)17-s + (3.12 − 6.53i)18-s − 2.82i·19-s + ⋯
L(s)  = 1  + (−1.20 − 1.20i)2-s + (0.816 + 0.577i)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 + (−1.10 + 1.56i)12-s + (0.554 + 0.554i)13-s − 0.534·14-s − 0.749·16-s + (0.685 + 0.685i)17-s + (0.735 − 1.54i)18-s − 0.648i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.998 + 0.0618i$
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: \((2,\ 825,\ (\ :1/2),\ 0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10535 - 0.0342246i\)
\(L(\frac12)\) \(\approx\) \(1.10535 - 0.0342246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 - i)T \)
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

−10.03686285230065771101216819356, −9.497279800979842560502145694141, −8.777028968383062239699382414313, −8.047120031833400554301041940332, −7.39515131259821647017444945482, −5.77662101769641470543053756919, −4.22074830615998325008382276256, −3.52040136551838228384131183023, −2.43294044064145393066262610440, −1.34696211033206194750660556907, 0.842624576763424979892029805874, 2.23183340527645237481113516078, 3.75751367762212821207982499877, 5.36942312791163025134934762212, 6.23017491468685556575326528040, 7.06790303383958444493729028924, 7.84371034881842602303204559406, 8.351257927010244544395782695325, 9.075142387860273600143080679798, 9.900126564152710241609596348682

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