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Properties

Label 2-4608-16.5-c1-0-44
Degree $2$
Conductor $4608$
Sign $0.707 + 0.707i$
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  + (0.414 + 0.414i)5-s + 3.69i·7-s + (−2.93 − 2.93i)11-s + (−2.41 + 2.41i)13-s + 2.82·17-s + (4.46 − 4.46i)19-s − 6.75i·23-s − 4.65i·25-s + (−5.24 + 5.24i)29-s + 3.06·31-s + (−1.53 + 1.53i)35-s + (−6.41 − 6.41i)37-s − 4i·41-s + (−0.765 − 0.765i)43-s + 3.06·47-s + ⋯
L(s)  = 1  + (0.185 + 0.185i)5-s + 1.39i·7-s + (−0.883 − 0.883i)11-s + (−0.669 + 0.669i)13-s + 0.685·17-s + (1.02 − 1.02i)19-s − 1.40i·23-s − 0.931i·25-s + (−0.973 + 0.973i)29-s + 0.549·31-s + (−0.258 + 0.258i)35-s + (−1.05 − 1.05i)37-s − 0.624i·41-s + (−0.116 − 0.116i)43-s + 0.446·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.457748501\)
\(L(\frac12)\) \(\approx\) \(1.457748501\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.414 - 0.414i)T + 5iT^{2} \)
7 \( 1 - 3.69iT - 7T^{2} \)
11 \( 1 + (2.93 + 2.93i)T + 11iT^{2} \)
13 \( 1 + (2.41 - 2.41i)T - 13iT^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \)
23 \( 1 + 6.75iT - 23T^{2} \)
29 \( 1 + (5.24 - 5.24i)T - 29iT^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 + (6.41 + 6.41i)T + 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + (0.765 + 0.765i)T + 43iT^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + (3.24 + 3.24i)T + 53iT^{2} \)
59 \( 1 + (-0.765 - 0.765i)T + 59iT^{2} \)
61 \( 1 + (0.757 - 0.757i)T - 61iT^{2} \)
67 \( 1 + (-1.39 + 1.39i)T - 67iT^{2} \)
71 \( 1 - 8.02iT - 71T^{2} \)
73 \( 1 - 6.48iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (-9.68 + 9.68i)T - 83iT^{2} \)
89 \( 1 + 4.82iT - 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−8.395597927170313703683724479530, −7.48634072572198136504487720619, −6.77685423881540882878880565799, −5.90533556757064326402698331447, −5.33583935878823323905743002728, −4.75162762514076690589420498949, −3.44278370232028399581985600166, −2.66337967503140854416957333739, −2.10184230498567696993783739633, −0.45821407081926405887590847091, 0.981318543708285256240329210519, 1.90537814277342388703578281988, 3.21496414290937257495852961715, 3.74111679233574204489428483054, 4.89436553694580317634883458153, 5.27126335501723462181381389798, 6.19307921365242756731808049135, 7.38303443549967555879104829683, 7.54783085661190731885439108844, 8.031178414424578023153192170173

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