Properties

Label 2-40e2-40.29-c1-0-31
Degree $2$
Conductor $1600$
Sign $0.316 + 0.948i$
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  + 2·3-s + 9-s − 6i·11-s − 6i·17-s − 2i·19-s − 4·27-s − 12i·33-s − 6·41-s + 10·43-s + 7·49-s − 12i·51-s − 4i·57-s + 6i·59-s + 14·67-s − 2i·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.333·9-s − 1.80i·11-s − 1.45i·17-s − 0.458i·19-s − 0.769·27-s − 2.08i·33-s − 0.937·41-s + 1.52·43-s + 49-s − 1.68i·51-s − 0.529i·57-s + 0.781i·59-s + 1.71·67-s − 0.234i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.252996842\)
\(L(\frac12)\) \(\approx\) \(2.252996842\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
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

−9.043286683129683219459596029754, −8.591579832571673505528003084598, −7.82336287633584810161167813566, −7.02556548992560164162144608904, −5.96858853173716671788493034377, −5.14925602938534690765238908263, −3.89753377453048086960262528029, −3.08735359786260095590437152702, −2.43016051313926583327314117215, −0.74244896642002666158509946344, 1.71493873068809262695085591796, 2.42481699904604462119437761230, 3.64768968698289207823728794986, 4.29569294913590630432367766231, 5.41852473123892389915316951896, 6.49255958603362394750607504820, 7.38715341089691132133790223606, 8.019838620753455863560859146050, 8.730054488693157569360021493151, 9.534829036418252700257629357443

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