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Properties

Label 2-2368-37.36-c1-0-71
Degree $2$
Conductor $2368$
Sign $-0.657 + 0.753i$
Analytic cond. $18.9085$
Root an. cond. $4.34839$
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  + 2.79·3-s − 3.79i·5-s − 2·7-s + 4.79·9-s − 3.79·11-s + 0.791i·13-s − 10.5i·15-s + 1.58i·17-s − 7.58i·19-s − 5.58·21-s − 0.791i·23-s − 9.37·25-s + 4.99·27-s − 0.791i·29-s − 5.37i·31-s + ⋯
L(s)  = 1  + 1.61·3-s − 1.69i·5-s − 0.755·7-s + 1.59·9-s − 1.14·11-s + 0.219i·13-s − 2.73i·15-s + 0.383i·17-s − 1.73i·19-s − 1.21·21-s − 0.164i·23-s − 1.87·25-s + 0.962·27-s − 0.146i·29-s − 0.965i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.657 + 0.753i$
Analytic conductor: \(18.9085\)
Root analytic conductor: \(4.34839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.135162802\)
\(L(\frac12)\) \(\approx\) \(2.135162802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (4 - 4.58i)T \)
good3 \( 1 - 2.79T + 3T^{2} \)
5 \( 1 + 3.79iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 - 0.791iT - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 + 7.58iT - 19T^{2} \)
23 \( 1 + 0.791iT - 23T^{2} \)
29 \( 1 + 0.791iT - 29T^{2} \)
31 \( 1 + 5.37iT - 31T^{2} \)
41 \( 1 - 5.20T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + 7.58iT - 59T^{2} \)
61 \( 1 - 8.20iT - 61T^{2} \)
67 \( 1 - 7.37T + 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 4.41iT - 97T^{2} \)
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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

−8.718682020742154788453587652056, −8.131664731025099697561730800888, −7.52397521145450639807884286036, −6.48124805816375689959541242271, −5.28597592303877060713771939646, −4.61641554909603812503223250174, −3.74652266801868183118272949905, −2.76874914202589151159143031377, −1.96772242624668441046302056651, −0.53024619584692557849770158882, 1.96191210290417094245436604385, 2.81080279284915227828569898289, 3.25705491625013597938919224255, 3.92217155651008412186320171042, 5.42799649979192886927991131571, 6.41150368932250250680772871955, 7.12057995466516196225358823544, 7.83933476409230407025945932743, 8.222574532934674868974645967015, 9.447671436703243390174000094296

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