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Properties

Label 2-40e2-5.4-c1-0-14
Degree $2$
Conductor $1600$
Sign $0.894 + 0.447i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
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 + 2i·7-s + 2·9-s − 5·11-s − 5i·17-s + 5·19-s + 2·21-s + 6i·23-s − 5i·27-s + 4·29-s + 10·31-s + 5i·33-s − 10i·37-s + 5·41-s − 4i·43-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.755i·7-s + 0.666·9-s − 1.50·11-s − 1.21i·17-s + 1.14·19-s + 0.436·21-s + 1.25i·23-s − 0.962i·27-s + 0.742·29-s + 1.79·31-s + 0.870i·33-s − 1.64i·37-s + 0.780·41-s − 0.609i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.711558732\)
\(L(\frac12)\) \(\approx\) \(1.711558732\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 5iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - iT - 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−9.444920249324078363851924348847, −8.451591241654617797928958535046, −7.48827396405676716769625049755, −7.31292880784136462291527510761, −5.99446293720170483428410644915, −5.34149140931135853662442739545, −4.47884914177644198254298165904, −3.00848023143031987805206135530, −2.33720387336688582995098144612, −0.899040366512349458854508529172, 0.988459762065002097456050266730, 2.53837244369318263664053504632, 3.57852601168250932625819451055, 4.54194332334893634143718979972, 5.09053976100896970407975199078, 6.27821204934432876523972304711, 7.08781247594143050208155581985, 8.011790149183488762133875175113, 8.489964967267012127252391952130, 9.885719314496483631087440279456

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