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Properties

Label 2-2300-115.114-c2-0-43
Degree $2$
Conductor $2300$
Sign $0.855 + 0.517i$
Analytic cond. $62.6704$
Root an. cond. $7.91646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.96i·3-s − 6.97·7-s − 15.6·9-s + 10.2i·11-s + 10.1i·13-s + 27.0·17-s − 19.5i·19-s − 34.6i·21-s + (−12.2 − 19.4i)23-s − 32.8i·27-s − 44.8·29-s + 5.79·31-s − 51.0·33-s − 12.8·37-s − 50.1·39-s + ⋯
L(s)  = 1  + 1.65i·3-s − 0.996·7-s − 1.73·9-s + 0.935i·11-s + 0.777i·13-s + 1.58·17-s − 1.02i·19-s − 1.64i·21-s + (−0.534 − 0.845i)23-s − 1.21i·27-s − 1.54·29-s + 0.187·31-s − 1.54·33-s − 0.348·37-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(62.6704\)
Root analytic conductor: \(7.91646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1),\ 0.855 + 0.517i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4685258599\)
\(L(\frac12)\) \(\approx\) \(0.4685258599\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + (12.2 + 19.4i)T \)
good3 \( 1 - 4.96iT - 9T^{2} \)
7 \( 1 + 6.97T + 49T^{2} \)
11 \( 1 - 10.2iT - 121T^{2} \)
13 \( 1 - 10.1iT - 169T^{2} \)
17 \( 1 - 27.0T + 289T^{2} \)
19 \( 1 + 19.5iT - 361T^{2} \)
29 \( 1 + 44.8T + 841T^{2} \)
31 \( 1 - 5.79T + 961T^{2} \)
37 \( 1 + 12.8T + 1.36e3T^{2} \)
41 \( 1 + 41.8T + 1.68e3T^{2} \)
43 \( 1 + 63.2T + 1.84e3T^{2} \)
47 \( 1 + 68.4iT - 2.20e3T^{2} \)
53 \( 1 - 21.5T + 2.80e3T^{2} \)
59 \( 1 + 29.6T + 3.48e3T^{2} \)
61 \( 1 + 69.8iT - 3.72e3T^{2} \)
67 \( 1 - 85.0T + 4.48e3T^{2} \)
71 \( 1 + 14.4T + 5.04e3T^{2} \)
73 \( 1 - 1.89iT - 5.32e3T^{2} \)
79 \( 1 + 24.6iT - 6.24e3T^{2} \)
83 \( 1 - 157.T + 6.88e3T^{2} \)
89 \( 1 - 109. iT - 7.92e3T^{2} \)
97 \( 1 - 174.T + 9.40e3T^{2} \)
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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

−9.105381192433611262747338424298, −8.208822173282595037342833256004, −7.10448618622004882136400122951, −6.38529178653710179876808834750, −5.30144996422153259023940795131, −4.79354323010399489516274146832, −3.76159112759409951547046913371, −3.34036415129389998177234695908, −2.08951086833077948675968533160, −0.13192422281878466220981015760, 0.938829496512218723612580385519, 1.84510149924133527338309007249, 3.19422231421500607154791928982, 3.48849348090769144219704744446, 5.45820023542200193943847064549, 5.87226310419040687343412368228, 6.53036628372435154832455975492, 7.52130104461099646888349843683, 7.86975867950168143536836840787, 8.618748282907863955103806807621

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