Properties

Label 2-12e3-12.11-c1-0-25
Degree $2$
Conductor $1728$
Sign $i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  − 2.82i·5-s − 1.73i·7-s + 4.89·11-s + 13-s + 2.82i·17-s − 5.19i·19-s + 4.89·23-s − 3.00·25-s + 5.65i·29-s − 3.46i·31-s − 4.89·35-s + 37-s − 5.65i·41-s − 3.46i·43-s − 4.89·47-s + ⋯
L(s)  = 1  − 1.26i·5-s − 0.654i·7-s + 1.47·11-s + 0.277·13-s + 0.685i·17-s − 1.19i·19-s + 1.02·23-s − 0.600·25-s + 1.05i·29-s − 0.622i·31-s − 0.828·35-s + 0.164·37-s − 0.883i·41-s − 0.528i·43-s − 0.714·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854705209\)
\(L(\frac12)\) \(\approx\) \(1.854705209\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 1.73iT - 79T^{2} \)
83 \( 1 + 9.79T + 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−8.895285527451951755009018166687, −8.722894785048230225576398480519, −7.48427055886027262673591317967, −6.78223786931276240965669413526, −5.88042799431600256309065630045, −4.82231514509249154918294601857, −4.25742208313532796767268826147, −3.30478726462640934107527356211, −1.63517492241153718724265220868, −0.789782212133105080676238111218, 1.44622892923630363541304710342, 2.71807492369847361217513547802, 3.45274099738324577098361050550, 4.45147334543587499772078685744, 5.68931335684947911323347140921, 6.41317143934561923967195591340, 6.96080511207827257690301636282, 7.893339868737661312967550000737, 8.819444625732403951224664829238, 9.531345510963549863083126791032

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