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Properties

Label 2-40e2-5.4-c1-0-7
Degree $2$
Conductor $1600$
Sign $0.447 - 0.894i$
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  + 3·9-s + 6i·13-s + 2i·17-s − 10·29-s + 2i·37-s + 10·41-s + 7·49-s + 14i·53-s + 10·61-s + 6i·73-s + 9·81-s − 10·89-s + 18i·97-s + 2·101-s + 6·109-s + ⋯
L(s)  = 1  + 9-s + 1.66i·13-s + 0.485i·17-s − 1.85·29-s + 0.328i·37-s + 1.56·41-s + 49-s + 1.92i·53-s + 1.28·61-s + 0.702i·73-s + 81-s − 1.05·89-s + 1.82i·97-s + 0.199·101-s + 0.574·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658334805\)
\(L(\frac12)\) \(\approx\) \(1.658334805\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−9.379985788756507890248421986987, −9.016694792407669942144884550905, −7.81098404526705947035644607784, −7.16211540055356908696640317727, −6.42305597257860151217384376818, −5.47609993860249359551185515327, −4.29583604653231160094339269388, −3.90274935645140785862603538659, −2.34529288830870757534092297515, −1.37774851511004780680395899227, 0.68967941519025122210458517661, 2.09752032955479824021626459538, 3.28514445811830652857474085937, 4.16525659046008017832538891650, 5.24094691624264550546619660031, 5.87475744794244571478260991634, 7.07700643386195257397833728416, 7.57493599446690188721245822483, 8.400361364554562061744997916979, 9.421389577647872954165604973444

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