Properties

Label 2-3200-5.4-c1-0-21
Degree $2$
Conductor $3200$
Sign $0.447 - 0.894i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
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  + 2.41i·3-s − 0.828i·7-s − 2.82·9-s − 5.24·11-s − 5.65i·13-s − 0.171i·17-s + 1.58·19-s + 1.99·21-s − 4.82i·23-s + 0.414i·27-s + 8·29-s + 0.828·31-s − 12.6i·33-s + 7.65i·37-s + 13.6·39-s + ⋯
L(s)  = 1  + 1.39i·3-s − 0.313i·7-s − 0.942·9-s − 1.58·11-s − 1.56i·13-s − 0.0416i·17-s + 0.363·19-s + 0.436·21-s − 1.00i·23-s + 0.0797i·27-s + 1.48·29-s + 0.148·31-s − 2.20i·33-s + 1.25i·37-s + 2.18·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (2049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.573561096\)
\(L(\frac12)\) \(\approx\) \(1.573561096\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41iT - 3T^{2} \)
7 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + 5.24T + 11T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 + 0.171iT - 17T^{2} \)
19 \( 1 - 1.58T + 19T^{2} \)
23 \( 1 + 4.82iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - 0.828T + 31T^{2} \)
37 \( 1 - 7.65iT - 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 9.65iT - 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 2.75iT - 67T^{2} \)
71 \( 1 - 9.65T + 71T^{2} \)
73 \( 1 + 5.82iT - 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 1.24iT - 83T^{2} \)
89 \( 1 + 6.17T + 89T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−8.864397634245682150303722798086, −7.980328954507207595381959617025, −7.66916444501208227545627127408, −6.32595186564942816715541268597, −5.56132818329605838215511769342, −4.78945963628878229836538094404, −4.37796570950588149668113930921, −2.97135219300863248273948531392, −2.87260710518587221396619414197, −0.77955967243387585716311151967, 0.70762900335616452250646992932, 2.02698896702243112711522493667, 2.43362422814152774647991266828, 3.70228757176455576848050922442, 4.85755038359512127093859230172, 5.63087714976238310658672095417, 6.40118596675179491683494071484, 7.19283281870393667078837072609, 7.59438604661351827332896620175, 8.399205988583068362521587816140

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