Properties

Label 2-4600-5.4-c1-0-12
Degree $2$
Conductor $4600$
Sign $-0.894 - 0.447i$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
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  + 2.29i·3-s − 1.61i·7-s − 2.26·9-s − 2.79·11-s − 4.88i·13-s + 3.54i·17-s + 2.79·19-s + 3.71·21-s + i·23-s + 1.67i·27-s + 3.36·29-s − 4.84·31-s − 6.41i·33-s + 3.87i·37-s + 11.2·39-s + ⋯
L(s)  = 1  + 1.32i·3-s − 0.611i·7-s − 0.756·9-s − 0.841·11-s − 1.35i·13-s + 0.859i·17-s + 0.640·19-s + 0.810·21-s + 0.208i·23-s + 0.322i·27-s + 0.624·29-s − 0.870·31-s − 1.11i·33-s + 0.636i·37-s + 1.79·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.125337225\)
\(L(\frac12)\) \(\approx\) \(1.125337225\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 - 2.29iT - 3T^{2} \)
7 \( 1 + 1.61iT - 7T^{2} \)
11 \( 1 + 2.79T + 11T^{2} \)
13 \( 1 + 4.88iT - 13T^{2} \)
17 \( 1 - 3.54iT - 17T^{2} \)
19 \( 1 - 2.79T + 19T^{2} \)
29 \( 1 - 3.36T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 - 3.87iT - 37T^{2} \)
41 \( 1 + 0.327T + 41T^{2} \)
43 \( 1 - 5.38iT - 43T^{2} \)
47 \( 1 - 0.0121iT - 47T^{2} \)
53 \( 1 - 0.866iT - 53T^{2} \)
59 \( 1 + 2.96T + 59T^{2} \)
61 \( 1 + 7.29T + 61T^{2} \)
67 \( 1 - 6.94iT - 67T^{2} \)
71 \( 1 + 2.16T + 71T^{2} \)
73 \( 1 - 6.02iT - 73T^{2} \)
79 \( 1 + 6.50T + 79T^{2} \)
83 \( 1 - 7.88iT - 83T^{2} \)
89 \( 1 - 9.71T + 89T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−8.676691279004127631465036000806, −7.904182214488956606676597863167, −7.41063177313916756808673474522, −6.28018374971027179379741381585, −5.43438037979987586785040853670, −4.97204842797564383403903496117, −4.08559720036746945237901444015, −3.42389886805738782261799699717, −2.69208264437826901328369703911, −1.14823470356046305133745746349, 0.32639880268806939173996251056, 1.62136978557804193686578423208, 2.29668831547900638599572479695, 3.10251640569607887537206814164, 4.35043054540193119789288563592, 5.20581101976438886077993914329, 5.94629023593061486593002913493, 6.69316731493012397603906848194, 7.33656284199830328089725449773, 7.75446773825665972165113640335

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