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Properties

Label 2-2432-8.5-c1-0-7
Degree $2$
Conductor $2432$
Sign $i$
Analytic cond. $19.4196$
Root an. cond. $4.40676$
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  + 2.46i·3-s + 3.22i·5-s − 4.04·7-s − 3.09·9-s + 6.41i·11-s − 0.815i·13-s − 7.96·15-s − 2.62·17-s − i·19-s − 9.97i·21-s − 6.16·23-s − 5.41·25-s − 0.233i·27-s − 3.85i·29-s + 7.43·31-s + ⋯
L(s)  = 1  + 1.42i·3-s + 1.44i·5-s − 1.52·7-s − 1.03·9-s + 1.93i·11-s − 0.226i·13-s − 2.05·15-s − 0.635·17-s − 0.229i·19-s − 2.17i·21-s − 1.28·23-s − 1.08·25-s − 0.0449i·27-s − 0.716i·29-s + 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $i$
Analytic conductor: \(19.4196\)
Root analytic conductor: \(4.40676\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2432,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8129449743\)
\(L(\frac12)\) \(\approx\) \(0.8129449743\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + iT \)
good3 \( 1 - 2.46iT - 3T^{2} \)
5 \( 1 - 3.22iT - 5T^{2} \)
7 \( 1 + 4.04T + 7T^{2} \)
11 \( 1 - 6.41iT - 11T^{2} \)
13 \( 1 + 0.815iT - 13T^{2} \)
17 \( 1 + 2.62T + 17T^{2} \)
23 \( 1 + 6.16T + 23T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 - 7.43T + 31T^{2} \)
37 \( 1 - 5.13iT - 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 0.0837iT - 43T^{2} \)
47 \( 1 - 3.99T + 47T^{2} \)
53 \( 1 - 4.22iT - 53T^{2} \)
59 \( 1 - 8.53iT - 59T^{2} \)
61 \( 1 + 5.72iT - 61T^{2} \)
67 \( 1 - 11.4iT - 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 - 7.25T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 6.51iT - 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−9.654309012788028931656682759340, −9.264365253401270003655201855178, −7.915062556579002435363885417258, −7.00940069990705942713617875938, −6.51896542087460255850078733826, −5.69616437722014839485710053591, −4.36489112381430393095663051422, −4.07258970240602493451064327897, −2.94642044184724920267892714552, −2.42986749490249463427650544010, 0.32269431650503877609098630534, 0.976732974636620877916835594544, 2.26248154743866684085597414360, 3.31983266245397219977151432408, 4.27669213002676029860876877780, 5.61073805641412627617078576512, 6.12121037728056907544932694883, 6.64935376613234102729967799433, 7.73447937348962307622943485131, 8.366016718056471929942599472159

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