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Properties

Label 2-4608-16.13-c1-0-36
Degree $2$
Conductor $4608$
Sign $0.923 - 0.382i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  + (1 − i)5-s + (1 + i)13-s + 2·17-s + 3i·25-s + (3 + 3i)29-s + (−5 + 5i)37-s + 10i·41-s + 7·49-s + (9 − 9i)53-s + (−1 − i)61-s + 2·65-s − 6i·73-s + (2 − 2i)85-s + 16i·89-s − 8·97-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s + (0.277 + 0.277i)13-s + 0.485·17-s + 0.600i·25-s + (0.557 + 0.557i)29-s + (−0.821 + 0.821i)37-s + 1.56i·41-s + 49-s + (1.23 − 1.23i)53-s + (−0.128 − 0.128i)61-s + 0.248·65-s − 0.702i·73-s + (0.216 − 0.216i)85-s + 1.69i·89-s − 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.109611700\)
\(L(\frac12)\) \(\approx\) \(2.109611700\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-9 + 9i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (1 + i)T + 61iT^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−8.495990798371836682178950568105, −7.66481479287116849880675349808, −6.86882153217375179451994709921, −6.18132580012906941399980901513, −5.35194630713359224657037072365, −4.79940927341321286917146669003, −3.79806402014332806114181623906, −2.98295216442246247129436463320, −1.86768030491735816623022548208, −0.995432605139636778114986141510, 0.69348739381659912831829692192, 1.98154852064004869687644201864, 2.77162956573359561061413093783, 3.69588338593725878564570192262, 4.48973493160142592018787505122, 5.59013336230329036198595987274, 5.92745409209069382125638039820, 6.91438854938860765028197656955, 7.42131480778202259531747235694, 8.350729149007756589859324713126

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