Properties

Label 2-1232-1232.83-c0-0-1
Degree $2$
Conductor $1232$
Sign $0.100 + 0.994i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.951 − 0.309i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s − 0.999·18-s + (−0.587 + 0.809i)22-s − 1.61·23-s + (−0.587 + 0.809i)25-s − 0.999i·28-s + (0.278 − 1.76i)29-s i·32-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.951 − 0.309i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s − 0.999·18-s + (−0.587 + 0.809i)22-s − 1.61·23-s + (−0.587 + 0.809i)25-s − 0.999i·28-s + (0.278 − 1.76i)29-s i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6511467010\)
\(L(\frac12)\) \(\approx\) \(0.6511467010\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.951 + 0.309i)T^{2} \)
5 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.951 - 0.309i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.0489 + 0.309i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-1.26 + 1.26i)T - iT^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 - 0.309i)T^{2} \)
61 \( 1 + (0.587 - 0.809i)T^{2} \)
67 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
71 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.745149631023169869555397255260, −9.135025513444030756172616407734, −8.029878360886762815891007615180, −7.47451901027952399421972385690, −6.54979256389811062788129360924, −5.90253607668828589766633811633, −4.05384698038450703274810934243, −3.63663880643696082644406310147, −2.19149708204248676182715529827, −0.798839813504973478374971980620, 1.64155345463761609303613665032, 2.57639248917550033800943596734, 4.08161489722350164908864359297, 5.22914503492577148879516587024, 6.21617415968002057559155486577, 6.88378214072705860151327755749, 7.70113846122415444775398663230, 8.505955720342526902482732752515, 9.407296095266355721160227335253, 9.936441051772792128800911380315

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