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Properties

Label 2-1232-7.4-c1-0-27
Degree $2$
Conductor $1232$
Sign $0.548 + 0.836i$
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  + (−1.09 + 1.90i)3-s + (0.317 + 0.550i)5-s + (−0.317 − 2.62i)7-s + (−0.917 − 1.58i)9-s + (−0.5 + 0.866i)11-s − 1.80·13-s − 1.39·15-s + (1.41 − 2.45i)17-s + (−2.78 − 4.81i)19-s + (5.35 + 2.28i)21-s + (1.08 + 1.87i)23-s + (2.29 − 3.98i)25-s − 2.56·27-s − 10.4·29-s + (3.21 − 5.56i)31-s + ⋯
L(s)  = 1  + (−0.634 + 1.09i)3-s + (0.142 + 0.246i)5-s + (−0.120 − 0.992i)7-s + (−0.305 − 0.529i)9-s + (−0.150 + 0.261i)11-s − 0.499·13-s − 0.360·15-s + (0.343 − 0.595i)17-s + (−0.638 − 1.10i)19-s + (1.16 + 0.498i)21-s + (0.225 + 0.391i)23-s + (0.459 − 0.796i)25-s − 0.493·27-s − 1.93·29-s + (0.577 − 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.548 + 0.836i$
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: \((2,\ 1232,\ (\ :1/2),\ 0.548 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7868395148\)
\(L(\frac12)\) \(\approx\) \(0.7868395148\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.317 + 2.62i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (1.09 - 1.90i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.317 - 0.550i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.80T + 13T^{2} \)
17 \( 1 + (-1.41 + 2.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.78 + 4.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.08 - 1.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + (-3.21 + 5.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.03 + 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.53T + 41T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + (1.41 + 2.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.73 - 6.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.90 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.16 - 3.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.801 - 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.29T + 71T^{2} \)
73 \( 1 + (-7.99 + 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.38 - 4.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + (0.182 + 0.315i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.59T + 97T^{2} \)
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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

−9.627178305734275338296898923422, −9.223349141839863456222090583026, −7.74680482786291212486948633758, −7.16782311631672208544716798966, −6.14163796704672907814091660834, −5.15988863424727815837787932123, −4.46337869524812666635360781589, −3.68847433820441268372841933806, −2.36409738056823928194155843305, −0.37538021015001455109964846054, 1.32982730852359801522948064659, 2.31038500703093615921506310929, 3.63134346634855261791962627160, 5.08866646091360890062128942053, 5.79156870555641851001475946957, 6.39695998192599771060681980240, 7.33613917270622166159249129063, 8.134533212467884322098856496594, 8.935015556536195813518089093765, 9.793608668008031491182825928370

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