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Properties

Label 2-48e2-48.35-c1-0-4
Degree $2$
Conductor $2304$
Sign $0.220 - 0.975i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  + (−1.16 − 1.16i)5-s − 3.74·7-s + (1.64 − 1.64i)11-s + (−0.645 − 0.645i)13-s + 6.57i·17-s + (1.41 − 1.41i)19-s − 6i·23-s − 2.29i·25-s + (−5.40 + 5.40i)29-s − 0.913i·31-s + (4.35 + 4.35i)35-s + (1 − i)37-s − 6.57·41-s + (3.74 + 3.74i)43-s − 6·47-s + ⋯
L(s)  = 1  + (−0.520 − 0.520i)5-s − 1.41·7-s + (0.496 − 0.496i)11-s + (−0.179 − 0.179i)13-s + 1.59i·17-s + (0.324 − 0.324i)19-s − 1.25i·23-s − 0.458i·25-s + (−1.00 + 1.00i)29-s − 0.164i·31-s + (0.736 + 0.736i)35-s + (0.164 − 0.164i)37-s − 1.02·41-s + (0.570 + 0.570i)43-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7180280916\)
\(L(\frac12)\) \(\approx\) \(0.7180280916\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 + 1.16i)T + 5iT^{2} \)
7 \( 1 + 3.74T + 7T^{2} \)
11 \( 1 + (-1.64 + 1.64i)T - 11iT^{2} \)
13 \( 1 + (0.645 + 0.645i)T + 13iT^{2} \)
17 \( 1 - 6.57iT - 17T^{2} \)
19 \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (5.40 - 5.40i)T - 29iT^{2} \)
31 \( 1 + 0.913iT - 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 6.57T + 41T^{2} \)
43 \( 1 + (-3.74 - 3.74i)T + 43iT^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-7.73 - 7.73i)T + 53iT^{2} \)
59 \( 1 + (9.29 - 9.29i)T - 59iT^{2} \)
61 \( 1 + (-10.2 - 10.2i)T + 61iT^{2} \)
67 \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \)
71 \( 1 + 2.70iT - 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \)
89 \( 1 - 0.412T + 89T^{2} \)
97 \( 1 - 2.70T + 97T^{2} \)
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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

−9.007450378921970565752010659700, −8.566088470710223106382874758765, −7.69221952801656483111714488672, −6.67485075102440878684778020257, −6.21510358898932630238759801321, −5.27022427304942809776602893795, −4.10831786576873789455243398737, −3.59063489498081554926650815722, −2.53595410088185795265573064952, −0.985499078238043068749854337629, 0.29638699504007341775799993802, 2.03932767706566540067926902341, 3.30456328648720018945393381648, 3.59922370599093055423352886501, 4.85540646258404095851690520789, 5.76881233967402122991771494654, 6.75151465700508210746121281998, 7.15312661028973864421584584256, 7.86301013715320840724527696917, 9.112294337593318647695501721860

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