Properties

Label 2-3200-40.29-c1-0-0
Degree $2$
Conductor $3200$
Sign $-0.894 - 0.447i$
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  + 0.317·3-s − 2.89·9-s − 3.78i·11-s − 1.89i·17-s + 5.97i·19-s − 1.87·27-s − 1.20i·33-s − 6.79·41-s − 8.48·43-s + 7·49-s − 0.603i·51-s + 1.89i·57-s + 14.1i·59-s − 16.3·67-s + 15.6i·73-s + ⋯
L(s)  = 1  + 0.183·3-s − 0.966·9-s − 1.14i·11-s − 0.460i·17-s + 1.37i·19-s − 0.360·27-s − 0.209i·33-s − 1.06·41-s − 1.29·43-s + 49-s − 0.0845i·51-s + 0.251i·57-s + 1.84i·59-s − 1.99·67-s + 1.83i·73-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2317442926\)
\(L(\frac12)\) \(\approx\) \(0.2317442926\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.317T + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 1.89iT - 17T^{2} \)
19 \( 1 - 5.97iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 16.3T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + 4.10T + 89T^{2} \)
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

−8.681703893546581773666194276250, −8.463416211059999353741812491823, −7.61091142923431111636320411084, −6.70225012914020184956702911575, −5.77690416367500846876460013364, −5.46171770321994109174046145133, −4.21136351492394598985681736918, −3.34091203388788965352702552212, −2.67591513594408314304671028176, −1.37070245846762250700265940440, 0.06695356096495571207658588099, 1.71312540201654906595398209230, 2.63692331324075320031265688784, 3.49316291295560357471349108757, 4.57945231574847202829901531909, 5.16967276121894812213395564487, 6.14700381639684445313610324206, 6.89008148694733605725965057351, 7.58877683597671246775380799231, 8.460602547901092457342711849296

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