Properties

Label 2-2352-12.11-c1-0-75
Degree $2$
Conductor $2352$
Sign $-0.998 + 0.0581i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (−0.777 + 1.54i)3-s − 3.40i·5-s + (−1.79 − 2.40i)9-s − 3.94·13-s + (5.26 + 2.64i)15-s + 6.09i·17-s − 4.38i·19-s + 8.33·23-s − 6.58·25-s + (5.11 − 0.901i)27-s − 3.80i·29-s − 4.89i·31-s − 7.58·37-s + (3.06 − 6.10i)39-s − 0.710i·41-s + ⋯
L(s)  = 1  + (−0.448 + 0.893i)3-s − 1.52i·5-s + (−0.597 − 0.802i)9-s − 1.09·13-s + (1.36 + 0.683i)15-s + 1.47i·17-s − 1.00i·19-s + 1.73·23-s − 1.31·25-s + (0.984 − 0.173i)27-s − 0.707i·29-s − 0.879i·31-s − 1.24·37-s + (0.491 − 0.978i)39-s − 0.110i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.998 + 0.0581i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.998 + 0.0581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1400207776\)
\(L(\frac12)\) \(\approx\) \(0.1400207776\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.777 - 1.54i)T \)
7 \( 1 \)
good5 \( 1 + 3.40iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 3.94T + 13T^{2} \)
17 \( 1 - 6.09iT - 17T^{2} \)
19 \( 1 + 4.38iT - 19T^{2} \)
23 \( 1 - 8.33T + 23T^{2} \)
29 \( 1 + 3.80iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + 7.58T + 37T^{2} \)
41 \( 1 + 0.710iT - 41T^{2} \)
43 \( 1 - 9.66iT - 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 + 4.66T + 59T^{2} \)
61 \( 1 + 1.70T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 6.59T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 6.09iT - 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−8.711958950549484455391953881400, −8.139495019604018336590420731851, −6.99492025473507840487022290824, −6.04000161236442355825431080525, −5.15831409102842920148404802238, −4.74850955216943236714950784959, −4.01807698845406068660930092428, −2.82666257941115630552499677494, −1.33574688500034873458905242142, −0.05127888894417966603979022133, 1.63441506154747344285367609335, 2.77399676771779220546015175981, 3.23239423167033405736791865589, 4.87233107934363217608942191435, 5.47612525214657235794339733349, 6.57596634571362579503972281288, 7.14206608412915332928609615029, 7.33626300063686018798684632370, 8.442003842453120843480590074108, 9.414787782050968279701357086906

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