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Properties

Label 2-15e2-15.14-c2-0-10
Degree $2$
Conductor $225$
Sign $-0.151 + 0.988i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 1.99·4-s − 9i·7-s − 8.48·8-s − 18.3i·11-s − i·13-s − 12.7i·14-s − 4.00·16-s + 26.8·17-s − 25·19-s − 26i·22-s + 4.24·23-s − 1.41i·26-s + 17.9i·28-s − 26.8i·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.499·4-s − 1.28i·7-s − 1.06·8-s − 1.67i·11-s − 0.0769i·13-s − 0.909i·14-s − 0.250·16-s + 1.58·17-s − 1.31·19-s − 1.18i·22-s + 0.184·23-s − 0.0543i·26-s + 0.642i·28-s − 0.926i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ -0.151 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.960352 - 1.11847i\)
\(L(\frac12)\) \(\approx\) \(0.960352 - 1.11847i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 1.41T + 4T^{2} \)
7 \( 1 + 9iT - 49T^{2} \)
11 \( 1 + 18.3iT - 121T^{2} \)
13 \( 1 + iT - 169T^{2} \)
17 \( 1 - 26.8T + 289T^{2} \)
19 \( 1 + 25T + 361T^{2} \)
23 \( 1 - 4.24T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 + 39T + 961T^{2} \)
37 \( 1 - 32iT - 1.36e3T^{2} \)
41 \( 1 - 5.65iT - 1.68e3T^{2} \)
43 \( 1 - 23iT - 1.84e3T^{2} \)
47 \( 1 + 32.5T + 2.20e3T^{2} \)
53 \( 1 - 96.1T + 2.80e3T^{2} \)
59 \( 1 - 9.89iT - 3.48e3T^{2} \)
61 \( 1 - 73T + 3.72e3T^{2} \)
67 \( 1 - 63iT - 4.48e3T^{2} \)
71 \( 1 + 62.2iT - 5.04e3T^{2} \)
73 \( 1 + 136iT - 5.32e3T^{2} \)
79 \( 1 - 24T + 6.24e3T^{2} \)
83 \( 1 - 46.6T + 6.88e3T^{2} \)
89 \( 1 - 101. iT - 7.92e3T^{2} \)
97 \( 1 + 7iT - 9.40e3T^{2} \)
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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

−11.86735927459109730195529965279, −10.83342756259383364404040163953, −9.949590094232477024298627650169, −8.701898940307724008625165957281, −7.81161061384981092440207150973, −6.36870648191100473249354892716, −5.41126079327799684409229033925, −4.08260669534730046180297217563, −3.29010064785278608502816317874, −0.66203721288307329801288480067, 2.22808630141681842520291173513, 3.75909257093194343656265699079, 5.01628279795112691035567279412, 5.74713748635186946928795010304, 7.12274354305100634448279968914, 8.504353338432426933027634306887, 9.315750499762115073302712231206, 10.22947812436824859330687022172, 11.74572996003332193039895952265, 12.65332991409113008401390650395

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