Properties

Label 2-1200-15.14-c2-0-26
Degree $2$
Conductor $1200$
Sign $-0.0806 - 0.996i$
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  + (1.55 + 2.56i)3-s + 1.34i·7-s + (−4.17 + 7.97i)9-s − 3.66i·11-s − 7.34i·13-s + 6.31·17-s + 30.7·19-s + (−3.44 + 2.08i)21-s + 26.2·23-s + (−26.9 + 1.68i)27-s + 40.9i·29-s + 9.97·31-s + (9.40 − 5.69i)33-s + 39.4i·37-s + (18.8 − 11.4i)39-s + ⋯
L(s)  = 1  + (0.517 + 0.855i)3-s + 0.191i·7-s + (−0.463 + 0.886i)9-s − 0.333i·11-s − 0.564i·13-s + 0.371·17-s + 1.61·19-s + (−0.164 + 0.0993i)21-s + 1.14·23-s + (−0.998 + 0.0624i)27-s + 1.41i·29-s + 0.321·31-s + (0.285 − 0.172i)33-s + 1.06i·37-s + (0.483 − 0.292i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0806 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0806 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0806 - 0.996i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.0806 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.279437379\)
\(L(\frac12)\) \(\approx\) \(2.279437379\)
\(L(2)\) 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 \( 1 + (-1.55 - 2.56i)T \)
5 \( 1 \)
good7 \( 1 - 1.34iT - 49T^{2} \)
11 \( 1 + 3.66iT - 121T^{2} \)
13 \( 1 + 7.34iT - 169T^{2} \)
17 \( 1 - 6.31T + 289T^{2} \)
19 \( 1 - 30.7T + 361T^{2} \)
23 \( 1 - 26.2T + 529T^{2} \)
29 \( 1 - 40.9iT - 841T^{2} \)
31 \( 1 - 9.97T + 961T^{2} \)
37 \( 1 - 39.4iT - 1.36e3T^{2} \)
41 \( 1 - 68.1iT - 1.68e3T^{2} \)
43 \( 1 - 24.0iT - 1.84e3T^{2} \)
47 \( 1 + 79.7T + 2.20e3T^{2} \)
53 \( 1 + 38.9T + 2.80e3T^{2} \)
59 \( 1 - 39.9iT - 3.48e3T^{2} \)
61 \( 1 + 64.2T + 3.72e3T^{2} \)
67 \( 1 + 53.9iT - 4.48e3T^{2} \)
71 \( 1 + 136. iT - 5.04e3T^{2} \)
73 \( 1 - 74.6iT - 5.32e3T^{2} \)
79 \( 1 - 79.6T + 6.24e3T^{2} \)
83 \( 1 - 0.0506T + 6.88e3T^{2} \)
89 \( 1 - 22.6iT - 7.92e3T^{2} \)
97 \( 1 - 35.0iT - 9.40e3T^{2} \)
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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

−9.658416326864753502648714217384, −9.108292114000202233255871005221, −8.182306699379505581911640089419, −7.56584352022837064239920910527, −6.36740974736341708768535916650, −5.24440939854702655183571555627, −4.78765531370897497825568890198, −3.26615545182295544918216910881, −3.03111324061963720957316589225, −1.27055820101116393647253637701, 0.69654892830893169624705314828, 1.85971577623307949472938704929, 2.96051223891873799296734418761, 3.91272881126656351292824721252, 5.16223332853322889708077100223, 6.11254112201448733470975424859, 7.14263037920483933689731585815, 7.49730188341628593804664609428, 8.489165393628687658839318097542, 9.308311101957422348661710431676

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