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Properties

Label 2-1232-7.2-c1-0-0
Degree $2$
Conductor $1232$
Sign $0.181 - 0.983i$
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  + (−1.29 − 2.24i)3-s + (−1.07 + 1.86i)5-s + (−0.726 − 2.54i)7-s + (−1.87 + 3.24i)9-s + (−0.5 − 0.866i)11-s − 0.0931·13-s + 5.58·15-s + (−3.02 − 5.23i)17-s + (−0.679 + 1.17i)19-s + (−4.77 + 4.93i)21-s + (−1.83 + 3.17i)23-s + (0.191 + 0.331i)25-s + 1.93·27-s + 0.458·29-s + (−0.118 − 0.205i)31-s + ⋯
L(s)  = 1  + (−0.749 − 1.29i)3-s + (−0.480 + 0.832i)5-s + (−0.274 − 0.961i)7-s + (−0.624 + 1.08i)9-s + (−0.150 − 0.261i)11-s − 0.0258·13-s + 1.44·15-s + (−0.733 − 1.27i)17-s + (−0.155 + 0.270i)19-s + (−1.04 + 1.07i)21-s + (−0.381 + 0.661i)23-s + (0.0382 + 0.0662i)25-s + 0.373·27-s + 0.0851·29-s + (−0.0213 − 0.0369i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.181 - 0.983i$
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: \((2,\ 1232,\ (\ :1/2),\ 0.181 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2297422787\)
\(L(\frac12)\) \(\approx\) \(0.2297422787\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.726 + 2.54i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (1.29 + 2.24i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.07 - 1.86i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.0931T + 13T^{2} \)
17 \( 1 + (3.02 + 5.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.679 - 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.83 - 3.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.458T + 29T^{2} \)
31 \( 1 + (0.118 + 0.205i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.78 - 8.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 + 2.67T + 43T^{2} \)
47 \( 1 + (1.02 - 1.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.93 - 12.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.27 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.01 + 3.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.50 + 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.60T + 71T^{2} \)
73 \( 1 + (4.00 + 6.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.85T + 83T^{2} \)
89 \( 1 + (1.28 - 2.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.990T + 97T^{2} \)
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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

−10.15890484188391972542875852984, −9.020897333122324581708516485846, −7.76106659317082132292757898079, −7.35214687790430070201730061707, −6.72853785086171844928176922840, −6.04712809409619691554819653727, −4.88908570151028663464640838483, −3.67933377859630644310708983854, −2.60580772615676626826678013121, −1.15693304858233229463536240767, 0.11924936113739282805221906206, 2.21128584986943005656922349033, 3.74904378889722250663619220813, 4.39903773544472343561177979296, 5.22162040880289154940363011214, 5.87624260331619605248630691179, 6.86206944420981954518681457229, 8.400060533604997034165902822514, 8.664173851263233652940154535576, 9.640287315785586071124631422483

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