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Properties

Label 2-48e2-48.35-c1-0-19
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  + (−0.517 − 0.517i)5-s − 1.41·7-s + (2.73 − 2.73i)11-s + (1.73 + 1.73i)13-s − 0.378i·17-s + (−0.378 + 0.378i)19-s − 3.46i·23-s − 4.46i·25-s + (−4.76 + 4.76i)29-s − 0.656i·31-s + (0.732 + 0.732i)35-s + (−4.46 + 4.46i)37-s + 10.1·41-s + (−6.31 − 6.31i)43-s + 10.3·47-s + ⋯
L(s)  = 1  + (−0.231 − 0.231i)5-s − 0.534·7-s + (0.823 − 0.823i)11-s + (0.480 + 0.480i)13-s − 0.0919i·17-s + (−0.0869 + 0.0869i)19-s − 0.722i·23-s − 0.892i·25-s + (−0.883 + 0.883i)29-s − 0.117i·31-s + (0.123 + 0.123i)35-s + (−0.733 + 0.733i)37-s + 1.58·41-s + (−0.962 − 0.962i)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: $\chi_{2304} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.0390 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320859954\)
\(L(\frac12)\) \(\approx\) \(1.320859954\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.517 + 0.517i)T + 5iT^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + (-2.73 + 2.73i)T - 11iT^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + 0.378iT - 17T^{2} \)
19 \( 1 + (0.378 - 0.378i)T - 19iT^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + (4.76 - 4.76i)T - 29iT^{2} \)
31 \( 1 + 0.656iT - 31T^{2} \)
37 \( 1 + (4.46 - 4.46i)T - 37iT^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + (6.31 + 6.31i)T + 43iT^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + (4.00 + 4.00i)T + 53iT^{2} \)
59 \( 1 + (-4.92 + 4.92i)T - 59iT^{2} \)
61 \( 1 + (-3 - 3i)T + 61iT^{2} \)
67 \( 1 + (1.03 - 1.03i)T - 67iT^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + 8.92iT - 73T^{2} \)
79 \( 1 + 11.9iT - 79T^{2} \)
83 \( 1 + (7.26 + 7.26i)T + 83iT^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 2.39T + 97T^{2} \)
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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

−8.841238703009196417064081771120, −8.226806866206243635334284222838, −7.17446566812089713538533085122, −6.44136719413557227394389581545, −5.84855190489394067299079725392, −4.72783814068788798513664097782, −3.86374330963046163264406552577, −3.14984867721440888342653711877, −1.81685007596041001294487925638, −0.49313189180522545684037689314, 1.23580200964198419393432036259, 2.47224732705584523056751745979, 3.60458974375386289454254272459, 4.13675897974009631469441292385, 5.36134014698184747032814538928, 6.08363509757879359033718586995, 6.97982219014001730747623460999, 7.51456539674404458706879868264, 8.427790420834234769132286969133, 9.420303171244242851198917848875

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