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Properties

Label 2-1100-5.4-c5-0-46
Degree $2$
Conductor $1100$
Sign $0.894 + 0.447i$
Analytic cond. $176.422$
Root an. cond. $13.2824$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.3i·3-s − 150. i·7-s − 397.·9-s + 121·11-s + 902. i·13-s − 815. i·17-s − 551.·19-s + 3.80e3·21-s + 242. i·23-s − 3.89e3i·27-s − 4.76e3·29-s − 3.32e3·31-s + 3.06e3i·33-s + 1.23e3i·37-s − 2.28e4·39-s + ⋯
L(s)  = 1  + 1.62i·3-s − 1.16i·7-s − 1.63·9-s + 0.301·11-s + 1.48i·13-s − 0.684i·17-s − 0.350·19-s + 1.88·21-s + 0.0955i·23-s − 1.02i·27-s − 1.05·29-s − 0.620·31-s + 0.489i·33-s + 0.148i·37-s − 2.40·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(176.422\)
Root analytic conductor: \(13.2824\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.016373225\)
\(L(\frac12)\) \(\approx\) \(1.016373225\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - 121T \)
good3 \( 1 - 25.3iT - 243T^{2} \)
7 \( 1 + 150. iT - 1.68e4T^{2} \)
13 \( 1 - 902. iT - 3.71e5T^{2} \)
17 \( 1 + 815. iT - 1.41e6T^{2} \)
19 \( 1 + 551.T + 2.47e6T^{2} \)
23 \( 1 - 242. iT - 6.43e6T^{2} \)
29 \( 1 + 4.76e3T + 2.05e7T^{2} \)
31 \( 1 + 3.32e3T + 2.86e7T^{2} \)
37 \( 1 - 1.23e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.30e3T + 1.15e8T^{2} \)
43 \( 1 - 1.51e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.04e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.58e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.04e4T + 7.14e8T^{2} \)
61 \( 1 - 2.48e4T + 8.44e8T^{2} \)
67 \( 1 + 5.24e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.33e4T + 1.80e9T^{2} \)
73 \( 1 + 7.65e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.39e4T + 3.07e9T^{2} \)
83 \( 1 + 6.45e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.00e4T + 5.58e9T^{2} \)
97 \( 1 - 1.53e5iT - 8.58e9T^{2} \)
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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

−9.431055448426011601879766343104, −8.507357776609470799238415924866, −7.34902208154298767037457033226, −6.57921267986606346314437806719, −5.39749716672431184430454668333, −4.44191719752494353487897351709, −4.06356325323100288475032986944, −3.17040513878674717636822549451, −1.70220009698406031003675085827, −0.22306254404440782742005653779, 0.830764163300624559113308027100, 1.90081171824099736616063626064, 2.56964607862071676541726678950, 3.69208207315286339063808531065, 5.45396919112555522870490369324, 5.76795638692153219261484402872, 6.75374797231138111917696780743, 7.50773987216750169429440783037, 8.377229196362982466929339851648, 8.762840782127558205148224077174

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