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Properties

Label 2-825-15.2-c1-0-12
Degree $2$
Conductor $825$
Sign $-0.391 - 0.920i$
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 + (−1.41 − i)3-s + 1.82i·4-s + (0.707 − 0.121i)6-s + (3.41 + 3.41i)7-s + (−1.12 − 1.12i)8-s + (1.00 + 2.82i)9-s + i·11-s + (1.82 − 2.58i)12-s + (2 − 2i)13-s − 2·14-s − 3·16-s + (−2.82 + 2.82i)17-s + (−1.12 − 0.535i)18-s − 2.82i·19-s + ⋯
L(s)  = 1  + (−0.207 + 0.207i)2-s + (−0.816 − 0.577i)3-s + 0.914i·4-s + (0.288 − 0.0495i)6-s + (1.29 + 1.29i)7-s + (−0.396 − 0.396i)8-s + (0.333 + 0.942i)9-s + 0.301i·11-s + (0.527 − 0.746i)12-s + (0.554 − 0.554i)13-s − 0.534·14-s − 0.750·16-s + (−0.685 + 0.685i)17-s + (−0.264 − 0.126i)18-s − 0.648i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.391 - 0.920i$
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: \((2,\ 825,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559245 + 0.845217i\)
\(L(\frac12)\) \(\approx\) \(0.559245 + 0.845217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 - 0.292i)T - 2iT^{2} \)
7 \( 1 + (-3.41 - 3.41i)T + 7iT^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (2.82 - 2.82i)T - 17iT^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + (-3.24 - 3.24i)T + 23iT^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (-0.171 - 0.171i)T + 37iT^{2} \)
41 \( 1 - 7.65iT - 41T^{2} \)
43 \( 1 + (0.242 - 0.242i)T - 43iT^{2} \)
47 \( 1 + (7.24 - 7.24i)T - 47iT^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 + (-6.41 - 6.41i)T + 67iT^{2} \)
71 \( 1 - 2.48iT - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 + (9.07 + 9.07i)T + 83iT^{2} \)
89 \( 1 - 9.65T + 89T^{2} \)
97 \( 1 + (10.6 + 10.6i)T + 97iT^{2} \)
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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

−10.91425113505337666255889579067, −9.415371067603548161626548408080, −8.501022986552727824103483335986, −8.021791799718282490655185084172, −7.13500561494779272224453310191, −6.16731434413030496508967783157, −5.27950012327672961362197637025, −4.40573890163771643283712766748, −2.79307672292130502550070732108, −1.63304861552945845191029180550, 0.62600548018974399612658021614, 1.75735816350239481435046658463, 3.80131330969066967900311546767, 4.68714575699672639251431252438, 5.32669152771374113443053634103, 6.44633939726195695453190438755, 7.18585303658935740544432848513, 8.478218723189264909730968647208, 9.293653019480754007015969552535, 10.25835523307782678635141197685

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