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Properties

Label 2-1232-7.4-c1-0-13
Degree $2$
Conductor $1232$
Sign $-0.421 - 0.906i$
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  + (−0.296 + 0.514i)3-s + (0.933 + 1.61i)5-s + (−0.0665 + 2.64i)7-s + (1.32 + 2.29i)9-s + (−0.5 + 0.866i)11-s + 4.32·13-s − 1.10·15-s + (−0.230 + 0.398i)17-s + (0.769 + 1.33i)19-s + (−1.33 − 0.819i)21-s + (−1.73 − 2.99i)23-s + (0.757 − 1.31i)25-s − 3.35·27-s − 5.78·29-s + (−0.487 + 0.844i)31-s + ⋯
L(s)  = 1  + (−0.171 + 0.296i)3-s + (0.417 + 0.723i)5-s + (−0.0251 + 0.999i)7-s + (0.441 + 0.764i)9-s + (−0.150 + 0.261i)11-s + 1.20·13-s − 0.286·15-s + (−0.0558 + 0.0967i)17-s + (0.176 + 0.305i)19-s + (−0.292 − 0.178i)21-s + (−0.360 − 0.624i)23-s + (0.151 − 0.262i)25-s − 0.645·27-s − 1.07·29-s + (−0.0875 + 0.151i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.421 - 0.906i$
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: \((2,\ 1232,\ (\ :1/2),\ -0.421 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.644333260\)
\(L(\frac12)\) \(\approx\) \(1.644333260\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0665 - 2.64i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.296 - 0.514i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.933 - 1.61i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.32T + 13T^{2} \)
17 \( 1 + (0.230 - 0.398i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.769 - 1.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + 2.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.78T + 29T^{2} \)
31 \( 1 + (0.487 - 0.844i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.133 - 0.230i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.485T + 41T^{2} \)
43 \( 1 - 3.70T + 43T^{2} \)
47 \( 1 + (-3.23 - 5.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.64 + 8.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.51 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.46T + 71T^{2} \)
73 \( 1 + (3.20 - 5.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.54 - 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.83T + 83T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.35T + 97T^{2} \)
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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

−9.983974422878940222358961781052, −9.285544311810762095593343702624, −8.354696728623244998818858675890, −7.58037552777078380708124562420, −6.44703456262121125804385164169, −5.87883281588945876080675385882, −4.96458761714490718307784340808, −3.88210641351482324329811659325, −2.70598951697083816369289414866, −1.77674510500064850294721690132, 0.75113211551770793046289809912, 1.65792698040462913562445642436, 3.45353559249725831980869933792, 4.13554697444229815571300062099, 5.32199693336945828300206705571, 6.11062866499012988151553871287, 6.98560974135041120895610171379, 7.73363777145684639083031259965, 8.766887434445903317490602140283, 9.401690401633672717544813354239

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