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Properties

Label 2-1232-7.4-c1-0-0
Degree $2$
Conductor $1232$
Sign $-0.198 + 0.980i$
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  + (−1.20 + 2.09i)3-s + (0.292 + 0.507i)5-s + (−1.62 + 2.09i)7-s + (−1.41 − 2.44i)9-s + (0.5 − 0.866i)11-s − 3.82·13-s − 1.41·15-s + (−1.82 + 3.16i)17-s + (−0.292 − 0.507i)19-s + (−2.41 − 5.91i)21-s + (−3.12 − 5.40i)23-s + (2.32 − 4.03i)25-s − 0.414·27-s + 2.65·29-s + (−2 + 3.46i)31-s + ⋯
L(s)  = 1  + (−0.696 + 1.20i)3-s + (0.130 + 0.226i)5-s + (−0.612 + 0.790i)7-s + (−0.471 − 0.816i)9-s + (0.150 − 0.261i)11-s − 1.06·13-s − 0.365·15-s + (−0.443 + 0.768i)17-s + (−0.0671 − 0.116i)19-s + (−0.526 − 1.29i)21-s + (−0.650 − 1.12i)23-s + (0.465 − 0.806i)25-s − 0.0797·27-s + 0.493·29-s + (−0.359 + 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.198 + 0.980i$
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: \((2,\ 1232,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1002383163\)
\(L(\frac12)\) \(\approx\) \(0.1002383163\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.09i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (1.20 - 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.292 - 0.507i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.292 + 0.507i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.70 - 8.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.41T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (5.24 + 9.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.94 - 6.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.79 - 4.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.91 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.37 + 2.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + (-4.70 + 8.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.62 + 11.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (6.24 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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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

−10.33976535336361511477772731819, −9.647374266896851547638980251922, −8.905676559841936710913831968250, −8.037247626090067700191743454572, −6.57415811418307677688208446974, −6.18855204999598024099276178240, −5.08140903004630123525536019630, −4.50281456137945828038769451159, −3.35806701696570337299925675011, −2.30616989307676701334637295369, 0.04769916397000864123673790202, 1.28972389525511750117094385503, 2.50964540095253481216939716324, 3.89926753677861942613400968106, 5.02463484823249842840674078748, 5.90937410513882109939019523014, 6.80526593034520889293757107562, 7.30365044964073052401500454316, 7.941170254620526812192221513244, 9.433911468092206656512134201754

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