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Properties

Label 2-1232-7.2-c1-0-35
Degree $2$
Conductor $1232$
Sign $0.276 + 0.961i$
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  + (0.956 + 1.65i)3-s + (1.78 − 3.09i)5-s + (−1.78 − 1.95i)7-s + (−0.328 + 0.568i)9-s + (−0.5 − 0.866i)11-s − 5.91·13-s + 6.82·15-s + (0.828 + 1.43i)17-s + (0.740 − 1.28i)19-s + (1.52 − 4.82i)21-s + (1.67 − 2.89i)23-s + (−3.86 − 6.70i)25-s + 4.48·27-s + 3.08·29-s + (−3.54 − 6.13i)31-s + ⋯
L(s)  = 1  + (0.552 + 0.956i)3-s + (0.798 − 1.38i)5-s + (−0.674 − 0.738i)7-s + (−0.109 + 0.189i)9-s + (−0.150 − 0.261i)11-s − 1.63·13-s + 1.76·15-s + (0.200 + 0.347i)17-s + (0.169 − 0.294i)19-s + (0.333 − 1.05i)21-s + (0.348 − 0.603i)23-s + (−0.773 − 1.34i)25-s + 0.862·27-s + 0.571·29-s + (−0.635 − 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.276 + 0.961i$
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: \((2,\ 1232,\ (\ :1/2),\ 0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.737542132\)
\(L(\frac12)\) \(\approx\) \(1.737542132\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 + 1.95i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.956 - 1.65i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.78 + 3.09i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.91T + 13T^{2} \)
17 \( 1 + (-0.828 - 1.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.740 + 1.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.67 + 2.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + (3.54 + 6.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.25 + 3.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 + 1.59T + 43T^{2} \)
47 \( 1 + (0.828 - 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.61 + 7.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.42 - 7.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.34 + 5.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.91 + 8.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.61T + 71T^{2} \)
73 \( 1 + (2.28 + 3.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.19 - 5.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.167T + 83T^{2} \)
89 \( 1 + (-1.28 + 2.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−9.644283762606481632888319580300, −9.017994533761011350294092913449, −8.155177425174895319418058932038, −7.13842493278175167567197651075, −6.05898906814755221773906480284, −4.99229139220925621255607040313, −4.50660678744246013946167748594, −3.46611672536704802273961941497, −2.29495902102269401203102613672, −0.65370761413572092905107085239, 1.82714383251163186580021518240, 2.66742907189736555975223689076, 3.13055333056054200192116533002, 4.98024820880859451275867713115, 5.89325825934370496199862025238, 6.98413915882863230600484833493, 7.05275298049361349063816843796, 8.080494228766291804247976950548, 9.190882718634672118683889687473, 9.913318358609430346953322907071

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