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Properties

Label 2-40e2-80.67-c1-0-2
Degree $2$
Conductor $1600$
Sign $-0.926 + 0.377i$
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  + 3.25i·3-s + (2.54 + 2.54i)7-s − 7.61·9-s + (0.462 + 0.462i)11-s − 1.33·13-s + (2.37 + 2.37i)17-s + (−2.69 − 2.69i)19-s + (−8.27 + 8.27i)21-s + (−2.10 + 2.10i)23-s − 15.0i·27-s + (−1.97 + 1.97i)29-s + 7.03i·31-s + (−1.50 + 1.50i)33-s − 7.81·37-s − 4.34i·39-s + ⋯
L(s)  = 1  + 1.88i·3-s + (0.960 + 0.960i)7-s − 2.53·9-s + (0.139 + 0.139i)11-s − 0.370·13-s + (0.575 + 0.575i)17-s + (−0.618 − 0.618i)19-s + (−1.80 + 1.80i)21-s + (−0.438 + 0.438i)23-s − 2.89i·27-s + (−0.367 + 0.367i)29-s + 1.26i·31-s + (−0.262 + 0.262i)33-s − 1.28·37-s − 0.696i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.926 + 0.377i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.321992770\)
\(L(\frac12)\) \(\approx\) \(1.321992770\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.25iT - 3T^{2} \)
7 \( 1 + (-2.54 - 2.54i)T + 7iT^{2} \)
11 \( 1 + (-0.462 - 0.462i)T + 11iT^{2} \)
13 \( 1 + 1.33T + 13T^{2} \)
17 \( 1 + (-2.37 - 2.37i)T + 17iT^{2} \)
19 \( 1 + (2.69 + 2.69i)T + 19iT^{2} \)
23 \( 1 + (2.10 - 2.10i)T - 23iT^{2} \)
29 \( 1 + (1.97 - 1.97i)T - 29iT^{2} \)
31 \( 1 - 7.03iT - 31T^{2} \)
37 \( 1 + 7.81T + 37T^{2} \)
41 \( 1 + 2.17iT - 41T^{2} \)
43 \( 1 - 3.10T + 43T^{2} \)
47 \( 1 + (0.0727 - 0.0727i)T - 47iT^{2} \)
53 \( 1 + 0.719iT - 53T^{2} \)
59 \( 1 + (8.67 - 8.67i)T - 59iT^{2} \)
61 \( 1 + (7.10 + 7.10i)T + 61iT^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-0.905 - 0.905i)T + 73iT^{2} \)
79 \( 1 - 3.90T + 79T^{2} \)
83 \( 1 + 6.02iT - 83T^{2} \)
89 \( 1 - 7.46T + 89T^{2} \)
97 \( 1 + (-3.74 - 3.74i)T + 97iT^{2} \)
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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

−9.833457658387613572603014555975, −9.028294657470123552218978223861, −8.638280584599861643268953442930, −7.76658126184382779273698987364, −6.35355639552022027366157358200, −5.29391670256095564023019583126, −5.05080777395704694477855343847, −4.05286568951556037993841932955, −3.18980418832337562327462798242, −2.03418719513437053954142335349, 0.50025072359381109252620764201, 1.56468508615092647396784931577, 2.40188652842457416880459711808, 3.73898001350900405158844308010, 4.95748776250507891199547324198, 5.94201062231114272892174405472, 6.68131204028631931805149778342, 7.54995974346504708338426285706, 7.84791946695714328444367541603, 8.570611271877433266636903148611

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