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Properties

Label 2-825-5.4-c1-0-13
Degree $2$
Conductor $825$
Sign $-0.447 + 0.894i$
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  − 2.76i·2-s + i·3-s − 5.62·4-s + 2.76·6-s + 1.86i·7-s + 10.0i·8-s − 9-s + 11-s − 5.62i·12-s − 4.62i·13-s + 5.14·14-s + 16.4·16-s + 2.49i·17-s + 2.76i·18-s + 5.38·19-s + ⋯
L(s)  = 1  − 1.95i·2-s + 0.577i·3-s − 2.81·4-s + 1.12·6-s + 0.704i·7-s + 3.54i·8-s − 0.333·9-s + 0.301·11-s − 1.62i·12-s − 1.28i·13-s + 1.37·14-s + 4.10·16-s + 0.604i·17-s + 0.650i·18-s + 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.655288 - 1.06027i\)
\(L(\frac12)\) \(\approx\) \(0.655288 - 1.06027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 2.76iT - 2T^{2} \)
7 \( 1 - 1.86iT - 7T^{2} \)
13 \( 1 + 4.62iT - 13T^{2} \)
17 \( 1 - 2.49iT - 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 + 7.14iT - 23T^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
31 \( 1 - 8.62T + 31T^{2} \)
37 \( 1 + 8.87iT - 37T^{2} \)
41 \( 1 + 0.761T + 41T^{2} \)
43 \( 1 - 7.40iT - 43T^{2} \)
47 \( 1 + 0.373iT - 47T^{2} \)
53 \( 1 - 5.45iT - 53T^{2} \)
59 \( 1 + 5.14T + 59T^{2} \)
61 \( 1 - 4.42T + 61T^{2} \)
67 \( 1 - 11.9iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 6.77iT - 73T^{2} \)
79 \( 1 - 6.01T + 79T^{2} \)
83 \( 1 + 14.5iT - 83T^{2} \)
89 \( 1 + 9.04T + 89T^{2} \)
97 \( 1 - 16.3iT - 97T^{2} \)
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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

−10.19865125312266161388746322097, −9.426014991221666121262555183727, −8.647549050252323649100020958192, −7.992333645045020627600034161995, −5.97018153005976497490813408474, −5.09527266976550820687975804697, −4.23043069662593306223666013062, −3.15688940903444291727880181408, −2.48770765736175355205274650945, −0.870957704152696615837526597776, 1.04798408301246304867224940283, 3.52121541720560906173723089509, 4.58765595344754486774577922356, 5.40650040803966792130461646539, 6.53313330763676062583773152006, 6.95285529812260908667459815091, 7.68928621327873034383018576407, 8.460134269320422693576210139323, 9.421434499251086336847146969236, 9.925360792011523267097625930993

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