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Properties

Label 2-1232-308.219-c1-0-8
Degree $2$
Conductor $1232$
Sign $-0.406 - 0.913i$
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  + (1.31 − 0.758i)3-s + (−0.222 + 0.385i)5-s + (−2.33 + 1.23i)7-s + (−0.350 + 0.607i)9-s + (−3.28 − 0.430i)11-s − 3.68i·13-s + 0.674i·15-s + (−1.86 + 1.07i)17-s + (−2.83 + 4.91i)19-s + (−2.13 + 3.39i)21-s + (7.10 + 4.10i)23-s + (2.40 + 4.15i)25-s + 5.61i·27-s + 4.09i·29-s + (−3.31 + 1.91i)31-s + ⋯
L(s)  = 1  + (0.758 − 0.437i)3-s + (−0.0994 + 0.172i)5-s + (−0.884 + 0.466i)7-s + (−0.116 + 0.202i)9-s + (−0.991 − 0.129i)11-s − 1.02i·13-s + 0.174i·15-s + (−0.452 + 0.261i)17-s + (−0.651 + 1.12i)19-s + (−0.466 + 0.740i)21-s + (1.48 + 0.854i)23-s + (0.480 + 0.831i)25-s + 1.07i·27-s + 0.760i·29-s + (−0.594 + 0.343i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.406 - 0.913i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.406 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9037699688\)
\(L(\frac12)\) \(\approx\) \(0.9037699688\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 - 1.23i)T \)
11 \( 1 + (3.28 + 0.430i)T \)
good3 \( 1 + (-1.31 + 0.758i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.222 - 0.385i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
17 \( 1 + (1.86 - 1.07i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.83 - 4.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.10 - 4.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.09iT - 29T^{2} \)
31 \( 1 + (3.31 - 1.91i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.522 + 0.904i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.32iT - 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (1.79 + 1.03i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.30 + 9.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.4 - 6.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 - 5.96i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.50 + 0.871i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (7.15 - 4.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.17 + 3.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + (-2.62 + 4.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.990397741771489211998233147499, −8.918044168076550565553786871188, −8.414830466774413594871183371350, −7.57154773801025126518804038523, −6.86514836804916883186777646481, −5.72234914904165258867494527425, −5.07418190749818849321980000398, −3.31067669874878128283173639388, −3.03789499291735361522691687923, −1.76316001067118033888722193928, 0.32221693987505327439324084919, 2.40946579591345948755856308111, 3.12770244924165650781748610825, 4.25552669268837163090318302184, 4.88514106578242340596120538766, 6.39940148588752890978266316787, 6.86335970598509270460497231854, 7.970017223308789652129640404160, 8.900152483447996805125817886868, 9.259718222645826488398217806844

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