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Properties

Label 2-1200-5.2-c2-0-16
Degree $2$
Conductor $1200$
Sign $0.991 + 0.130i$
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 + (−0.325 − 0.325i)7-s − 2.99i·9-s + 11.7·11-s + (3.67 − 3.67i)13-s + (11.3 + 11.3i)17-s + 30.3i·19-s − 0.797·21-s + (−9.55 + 9.55i)23-s + (−3.67 − 3.67i)27-s − 15.3i·29-s + 21.2·31-s + (14.4 − 14.4i)33-s + (3.10 + 3.10i)37-s − 9i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.0465 − 0.0465i)7-s − 0.333i·9-s + 1.07·11-s + (0.282 − 0.282i)13-s + (0.667 + 0.667i)17-s + 1.59i·19-s − 0.0379·21-s + (−0.415 + 0.415i)23-s + (−0.136 − 0.136i)27-s − 0.530i·29-s + 0.683·31-s + (0.437 − 0.437i)33-s + (0.0838 + 0.0838i)37-s − 0.230i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.496791448\)
\(L(\frac12)\) \(\approx\) \(2.496791448\)
\(L(2)\) 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 \( 1 + (0.325 + 0.325i)T + 49iT^{2} \)
11 \( 1 - 11.7T + 121T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 169iT^{2} \)
17 \( 1 + (-11.3 - 11.3i)T + 289iT^{2} \)
19 \( 1 - 30.3iT - 361T^{2} \)
23 \( 1 + (9.55 - 9.55i)T - 529iT^{2} \)
29 \( 1 + 15.3iT - 841T^{2} \)
31 \( 1 - 21.2T + 961T^{2} \)
37 \( 1 + (-3.10 - 3.10i)T + 1.36e3iT^{2} \)
41 \( 1 - 18.2T + 1.68e3T^{2} \)
43 \( 1 + (19.4 - 19.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (27.7 + 27.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-56.8 + 56.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 82iT - 3.48e3T^{2} \)
61 \( 1 - 94.5T + 3.72e3T^{2} \)
67 \( 1 + (-12.7 - 12.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 77.7T + 5.04e3T^{2} \)
73 \( 1 + (-90.2 + 90.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 103. iT - 6.24e3T^{2} \)
83 \( 1 + (22.8 - 22.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 159. iT - 7.92e3T^{2} \)
97 \( 1 + (56.7 + 56.7i)T + 9.40e3iT^{2} \)
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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

−9.715911908120126769218597802318, −8.406676682390925420062107526645, −8.162775474322339001740415202451, −7.05276349684135913224690004212, −6.24274651150057130257037224079, −5.49565131181975819346817833382, −4.00638054127094668878047331853, −3.48925901480941052106588363309, −2.04157745915290759940606017999, −1.04384470635856980988729038490, 0.911729691638754349029282091223, 2.37255062756539937161898123439, 3.39549932123022915379948786092, 4.34154644560772684971714290329, 5.16672051688202435136503043659, 6.35482594459534226370232423437, 7.04797005033444124409814495226, 8.051071274146432990653426861688, 8.972320259261984830985927889251, 9.389738977675522209027797389038

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