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Properties

Label 2-1232-7.4-c1-0-1
Degree $2$
Conductor $1232$
Sign $-0.832 - 0.553i$
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  + (−1.32 + 2.29i)3-s + (−1.82 − 3.15i)5-s + (−1.32 − 2.29i)7-s + (−2 − 3.46i)9-s + (0.5 − 0.866i)11-s + 5·13-s + 9.64·15-s + (−3 + 5.19i)17-s + (−0.177 − 0.306i)19-s + 7·21-s + (−1.82 − 3.15i)23-s + (−4.14 + 7.18i)25-s + 2.64·27-s − 4.29·29-s + (−2 + 3.46i)31-s + ⋯
L(s)  = 1  + (−0.763 + 1.32i)3-s + (−0.815 − 1.41i)5-s + (−0.499 − 0.866i)7-s + (−0.666 − 1.15i)9-s + (0.150 − 0.261i)11-s + 1.38·13-s + 2.49·15-s + (−0.727 + 1.26i)17-s + (−0.0406 − 0.0703i)19-s + 1.52·21-s + (−0.380 − 0.658i)23-s + (−0.829 + 1.43i)25-s + 0.509·27-s − 0.796·29-s + (−0.359 + 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2848268178\)
\(L(\frac12)\) \(\approx\) \(0.2848268178\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (1.32 - 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.82 + 3.15i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.177 + 0.306i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.82 + 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.29T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.822 - 1.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-6.64 - 11.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.82 + 3.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.322 - 0.559i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.85 + 3.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.96 + 3.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.64T + 71T^{2} \)
73 \( 1 + (2.82 - 4.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (-7.29 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−10.17360504432381111340473247458, −9.144080362816378399087266048205, −8.683718563709741498593963733658, −7.80900634996902077385335725779, −6.45594071446622869540642642156, −5.73312593564864172002434468158, −4.69593950748203889698440332178, −4.03249708460195424156216601753, −3.66002031534874213529928011374, −1.16351444443554253146833506058, 0.15241677916266985971503274949, 1.92763117877507553264899257053, 2.95769188044307850401737459651, 3.97445699120429548594873416099, 5.63040366250997921273855025475, 6.14776889474822562283330918514, 7.06815205032687798235384504273, 7.28347800109832380409893705846, 8.378912396142693594541879753035, 9.313318460678263851091680578203

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