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Properties

Label 2-15e2-9.4-c3-0-14
Degree $2$
Conductor $225$
Sign $0.0997 + 0.995i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.16 − 3.75i)2-s + (−0.193 + 5.19i)3-s + (−5.39 + 9.34i)4-s + (19.9 − 10.5i)6-s + (−6.50 − 11.2i)7-s + 12.1·8-s + (−26.9 − 2.01i)9-s + (17.2 + 29.9i)11-s + (−47.5 − 29.8i)12-s + (3.77 − 6.54i)13-s + (−28.1 + 48.8i)14-s + (16.9 + 29.2i)16-s − 82.0·17-s + (50.8 + 105. i)18-s + 146.·19-s + ⋯
L(s)  = 1  + (−0.766 − 1.32i)2-s + (−0.0373 + 0.999i)3-s + (−0.674 + 1.16i)4-s + (1.35 − 0.716i)6-s + (−0.351 − 0.607i)7-s + 0.535·8-s + (−0.997 − 0.0745i)9-s + (0.473 + 0.820i)11-s + (−1.14 − 0.717i)12-s + (0.0806 − 0.139i)13-s + (−0.538 + 0.931i)14-s + (0.264 + 0.457i)16-s − 1.17·17-s + (0.665 + 1.38i)18-s + 1.76·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0997 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0997 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.0997 + 0.995i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.0997 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.683778 - 0.618680i\)
\(L(\frac12)\) \(\approx\) \(0.683778 - 0.618680i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.193 - 5.19i)T \)
5 \( 1 \)
good2 \( 1 + (2.16 + 3.75i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (6.50 + 11.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-17.2 - 29.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-3.77 + 6.54i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 82.0T + 4.91e3T^{2} \)
19 \( 1 - 146.T + 6.85e3T^{2} \)
23 \( 1 + (-93.9 + 162. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (21.8 + 37.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-29.3 + 50.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 329.T + 5.06e4T^{2} \)
41 \( 1 + (-86.4 + 149. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-78.3 - 135. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-84.6 - 146. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 609.T + 1.48e5T^{2} \)
59 \( 1 + (-297. + 514. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (162. + 281. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-344. + 596. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 515.T + 3.57e5T^{2} \)
73 \( 1 - 1.08e3T + 3.89e5T^{2} \)
79 \( 1 + (-262. - 455. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-179. - 310. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.51e3T + 7.04e5T^{2} \)
97 \( 1 + (-197. - 342. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.19180182548546896843852370232, −10.66217933102895790519669181550, −9.524111548470367676781497237820, −9.369735460590859693545509392454, −8.010114456956847109242658410866, −6.49083535301710403819738657757, −4.74887608915904820629419673524, −3.70299164179570601901153681764, −2.52790187606086896147840763034, −0.65800697878304793198026705792, 0.999672267602924492798341047880, 3.01881461697562790687190688027, 5.40146085439402333411826415386, 6.15946976242015277679681677840, 7.06117262967820398186230592738, 7.86516019618231113038936145265, 8.954400518927496232259648470177, 9.387776611485774380544283546147, 11.21711760284167656782455440438, 11.94541021021411003418279694915

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