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Properties

Label 2-48e2-48.11-c1-0-27
Degree $2$
Conductor $2304$
Sign $0.0390 + 0.999i$
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  + (1.93 − 1.93i)5-s + 1.41·7-s + (0.732 + 0.732i)11-s + (1.73 − 1.73i)13-s − 5.27i·17-s + (−5.27 − 5.27i)19-s − 3.46i·23-s − 2.46i·25-s + (−2.31 − 2.31i)29-s + 9.14i·31-s + (2.73 − 2.73i)35-s + (−2.46 − 2.46i)37-s + 4.52·41-s + (3.48 − 3.48i)43-s − 10.3·47-s + ⋯
L(s)  = 1  + (0.863 − 0.863i)5-s + 0.534·7-s + (0.220 + 0.220i)11-s + (0.480 − 0.480i)13-s − 1.28i·17-s + (−1.21 − 1.21i)19-s − 0.722i·23-s − 0.492i·25-s + (−0.429 − 0.429i)29-s + 1.64i·31-s + (0.461 − 0.461i)35-s + (−0.405 − 0.405i)37-s + 0.705·41-s + (0.531 − 0.531i)43-s − 1.51·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0390 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0390 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.0390 + 0.999i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.0390 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.075436495\)
\(L(\frac12)\) \(\approx\) \(2.075436495\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 1.93i)T - 5iT^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + (-0.732 - 0.732i)T + 11iT^{2} \)
13 \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \)
17 \( 1 + 5.27iT - 17T^{2} \)
19 \( 1 + (5.27 + 5.27i)T + 19iT^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + (2.31 + 2.31i)T + 29iT^{2} \)
31 \( 1 - 9.14iT - 31T^{2} \)
37 \( 1 + (2.46 + 2.46i)T + 37iT^{2} \)
41 \( 1 - 4.52T + 41T^{2} \)
43 \( 1 + (-3.48 + 3.48i)T - 43iT^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + (-8.24 + 8.24i)T - 53iT^{2} \)
59 \( 1 + (-8.92 - 8.92i)T + 59iT^{2} \)
61 \( 1 + (3 - 3i)T - 61iT^{2} \)
67 \( 1 + (-3.86 - 3.86i)T + 67iT^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 + 4.92iT - 73T^{2} \)
79 \( 1 + 2.17iT - 79T^{2} \)
83 \( 1 + (-10.7 + 10.7i)T - 83iT^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−8.817398727712554563441887712997, −8.320766984827710156170771308584, −7.17327354034033126497676153024, −6.52130149711382261794965368300, −5.44537130075423064008167110116, −4.98175501675884148562991940401, −4.14454965845824365866587811362, −2.77858422860297857283926320515, −1.84553409624276569199902387916, −0.70077652915045827259313585044, 1.58585953695447658876286856088, 2.20546580515623896645885719592, 3.53741007635323264091487731875, 4.20495672333205832980266645605, 5.46936013685741126092995734641, 6.23380964045586769453821093020, 6.55088058383847766027303521012, 7.82461245807987811832855427619, 8.286690718016920471700518917126, 9.310754442786436994979400733453

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