Properties

Label 2-1200-15.14-c0-0-1
Degree $2$
Conductor $1200$
Sign $-0.447 + 0.894i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s i·7-s − 9-s i·13-s − 19-s − 21-s + i·27-s + 31-s − 2i·37-s − 39-s + i·43-s + i·57-s − 61-s + i·63-s i·67-s + ⋯
L(s)  = 1  i·3-s i·7-s − 9-s i·13-s − 19-s − 21-s + i·27-s + 31-s − 2i·37-s − 39-s + i·43-s + i·57-s − 61-s + i·63-s i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9236589007\)
\(L(\frac12)\) \(\approx\) \(0.9236589007\)
\(L(1)\) 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 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−9.712988721814415946608443269753, −8.637800860934582289020213754342, −7.915532844020275280223287750422, −7.28822144623843650417256240064, −6.44497921649453797621235043873, −5.65924444420486887252769112857, −4.48280741826464614963215358653, −3.36691859714621788828279753969, −2.22942052793984426763547218566, −0.817156158108819163320116772205, 2.10863678667909022249764221974, 3.14018841927764738210195200695, 4.28823171968519359752415878042, 4.96767717499576217254459612318, 5.99161771659389753301739223094, 6.66021730851117044885065126857, 8.076622587861965813696973815329, 8.760434694483442977948695514813, 9.324223472328552274411279417898, 10.16692126896792320217436702249

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