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Properties

Label 2-15e2-15.2-c1-0-2
Degree $2$
Conductor $225$
Sign $0.998 - 0.0618i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + (0.707 − 0.707i)2-s + 0.999i·4-s + (2 + 2i)7-s + (2.12 + 2.12i)8-s − 2.82i·11-s + (−1 + i)13-s + 2.82·14-s + 1.00·16-s + (2.82 − 2.82i)17-s + (−2.00 − 2.00i)22-s + (−2.82 − 2.82i)23-s + 1.41i·26-s + (−1.99 + 1.99i)28-s − 4.24·29-s − 4·31-s + (−3.53 + 3.53i)32-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + 0.499i·4-s + (0.755 + 0.755i)7-s + (0.750 + 0.750i)8-s − 0.852i·11-s + (−0.277 + 0.277i)13-s + 0.755·14-s + 0.250·16-s + (0.685 − 0.685i)17-s + (−0.426 − 0.426i)22-s + (−0.589 − 0.589i)23-s + 0.277i·26-s + (−0.377 + 0.377i)28-s − 0.787·29-s − 0.718·31-s + (−0.624 + 0.624i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.998 - 0.0618i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.998 - 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66462 + 0.0515409i\)
\(L(\frac12)\) \(\approx\) \(1.66462 + 0.0515409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-8 + 8i)T - 43iT^{2} \)
47 \( 1 + (5.65 - 5.65i)T - 47iT^{2} \)
53 \( 1 + (2.82 + 2.82i)T + 53iT^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 + (2.82 + 2.82i)T + 83iT^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-11 - 11i)T + 97iT^{2} \)
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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

−12.12361151186186338459818859337, −11.53934067007424217991385147819, −10.69460255020947430415038567067, −9.226469386394698919591401633144, −8.299540793805372021990532734293, −7.41439774919426906661234688227, −5.77736108683814380360010462353, −4.76327402091490315405806228426, −3.43575388767802316861762507108, −2.15141207381034754334356551336, 1.60237081858699516247308595620, 3.92599680435465843769319585591, 4.92528642053359984071928104512, 5.94259361664127967526430284693, 7.22408914352540503292374627314, 7.87402606288572129739644673693, 9.509366422982229982987214642022, 10.32011556253008692366435768901, 11.16369469681601332877654522516, 12.44350411956089834311056661805

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