Properties

Label 2-1232-7.2-c1-0-13
Degree $2$
Conductor $1232$
Sign $0.266 - 0.963i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (0.5 + 0.866i)3-s + (0.5 + 2.59i)7-s + (1 − 1.73i)9-s + (−0.5 − 0.866i)11-s − 13-s + (3 + 5.19i)17-s + (1 − 1.73i)19-s + (−2 + 1.73i)21-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s + 5·27-s + 9·29-s + (−2 − 3.46i)31-s + (0.499 − 0.866i)33-s + (−1 + 1.73i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.188 + 0.981i)7-s + (0.333 − 0.577i)9-s + (−0.150 − 0.261i)11-s − 0.277·13-s + (0.727 + 1.26i)17-s + (0.229 − 0.397i)19-s + (−0.436 + 0.377i)21-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s + 0.962·27-s + 1.67·29-s + (−0.359 − 0.622i)31-s + (0.0870 − 0.150i)33-s + (−0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842761952\)
\(L(\frac12)\) \(\approx\) \(1.842761952\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−9.815991239653955782599352048980, −9.090267695284946990880242048174, −8.412662959080484984361178184546, −7.57374038880333160460192713466, −6.43598085356043037046923057859, −5.66571763480823976660190355906, −4.76231918368796292041647115324, −3.67218872598461066213710518050, −2.85905336288879770823024735656, −1.45226463344471417784925838315, 0.828614476607184150786913457889, 2.13929464396301414031242018120, 3.21685769585488133189518104713, 4.52381749628072942167802289990, 5.06407446064972615015910222858, 6.53147148132224377359485810026, 7.12505369747214846911742528632, 7.897492564547425010964100935566, 8.460082282030837681224886975991, 9.746976082664709018941615809743

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