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Properties

Label 2-3225-129.92-c0-0-1
Degree $2$
Conductor $3225$
Sign $-0.216 + 0.976i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s − 0.999·21-s − 0.999·27-s + (−0.5 − 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)36-s + (−0.5 + 0.866i)37-s − 0.999·39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s − 0.999·21-s − 0.999·27-s + (−0.5 − 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)36-s + (−0.5 + 0.866i)37-s − 0.999·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (2801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ -0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.682366183\)
\(L(\frac12)\) \(\approx\) \(1.682366183\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
43 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 - T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.457050305970332728454482320213, −7.59606214608136902596130078616, −7.20521089444688835923754073434, −6.64396971635248739065318250474, −5.84574811856408765219938812136, −4.86543073037828156442217677859, −3.38683778928594834562929066361, −3.09429789678970438930244399705, −1.99920889551690723028689585704, −0.913155099896320989058739339493, 1.97838051063871661224911396413, 2.51898933279362679959435390369, 3.49964676172913062298886587011, 4.20352490169951064922443073703, 5.53309796727399889917376322423, 5.76558874085281601850522753858, 6.90610476231674821993839676812, 7.55540472707940254288964896764, 8.385095800075940724075746696647, 9.169490397562938606947072852035

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