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Properties

Label 2-1200-5.3-c2-0-27
Degree $2$
Conductor $1200$
Sign $0.326 + 0.945i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
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  + (1.22 + 1.22i)3-s + (−7.22 + 7.22i)7-s + 2.99i·9-s + 8.69·11-s + (−15.6 − 15.6i)13-s + (13.3 − 13.3i)17-s − 4.30i·19-s − 17.6·21-s + (−28.0 − 28.0i)23-s + (−3.67 + 3.67i)27-s + 20.6i·29-s − 39.0·31-s + (10.6 + 10.6i)33-s + (12.4 − 12.4i)37-s − 38.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−1.03 + 1.03i)7-s + 0.333i·9-s + 0.790·11-s + (−1.20 − 1.20i)13-s + (0.785 − 0.785i)17-s − 0.226i·19-s − 0.842·21-s + (−1.21 − 1.21i)23-s + (−0.136 + 0.136i)27-s + 0.713i·29-s − 1.26·31-s + (0.322 + 0.322i)33-s + (0.337 − 0.337i)37-s − 0.984i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.180018322\)
\(L(\frac12)\) \(\approx\) \(1.180018322\)
\(L(2)\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (7.22 - 7.22i)T - 49iT^{2} \)
11 \( 1 - 8.69T + 121T^{2} \)
13 \( 1 + (15.6 + 15.6i)T + 169iT^{2} \)
17 \( 1 + (-13.3 + 13.3i)T - 289iT^{2} \)
19 \( 1 + 4.30iT - 361T^{2} \)
23 \( 1 + (28.0 + 28.0i)T + 529iT^{2} \)
29 \( 1 - 20.6iT - 841T^{2} \)
31 \( 1 + 39.0T + 961T^{2} \)
37 \( 1 + (-12.4 + 12.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 62.6T + 1.68e3T^{2} \)
43 \( 1 + (-11.8 - 11.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-58.0 + 58.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-0.606 - 0.606i)T + 2.80e3iT^{2} \)
59 \( 1 + 30iT - 3.48e3T^{2} \)
61 \( 1 - 69.7T + 3.72e3T^{2} \)
67 \( 1 + (-5.02 + 5.02i)T - 4.48e3iT^{2} \)
71 \( 1 - 38.6T + 5.04e3T^{2} \)
73 \( 1 + (46.2 + 46.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 31.3iT - 6.24e3T^{2} \)
83 \( 1 + (39.4 + 39.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (-54.1 + 54.1i)T - 9.40e3iT^{2} \)
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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

−9.443144707430928419816027003384, −8.788000695785732324183505687809, −7.80102069512398490148557319891, −6.96646308043154752181297970304, −5.87679886585109379021266805905, −5.26059200042317281586202758491, −4.03658142746324149281189152541, −3.01299649611871756992015656577, −2.34404009596390475229535528528, −0.34531496486847060580837417062, 1.20226999584431711328225015330, 2.38898855904227218528840511692, 3.79801915471955881209178657080, 4.08503424591572153689399543148, 5.72905916004528641414694977266, 6.52248285267377988059035785870, 7.33651417661973991213980542801, 7.80062276168227624318715579298, 9.161615728403394254174231600504, 9.610893845877013264617228725774

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