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Properties

Label 2-221-13.12-c1-0-2
Degree $2$
Conductor $221$
Sign $-0.908 - 0.417i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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  + 1.36i·2-s − 1.68·3-s + 0.148·4-s + 1.63i·5-s − 2.29i·6-s − 0.615i·7-s + 2.92i·8-s − 0.148·9-s − 2.22·10-s + 5.26i·11-s − 0.250·12-s + (−3.27 − 1.50i)13-s + 0.836·14-s − 2.76i·15-s − 3.68·16-s + 17-s + ⋯
L(s)  = 1  + 0.962i·2-s − 0.974·3-s + 0.0741·4-s + 0.732i·5-s − 0.938i·6-s − 0.232i·7-s + 1.03i·8-s − 0.0494·9-s − 0.704·10-s + 1.58i·11-s − 0.0722·12-s + (−0.908 − 0.417i)13-s + 0.223·14-s − 0.714i·15-s − 0.920·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.908 - 0.417i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ -0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176148 + 0.805711i\)
\(L(\frac12)\) \(\approx\) \(0.176148 + 0.805711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.27 + 1.50i)T \)
17 \( 1 - T \)
good2 \( 1 - 1.36iT - 2T^{2} \)
3 \( 1 + 1.68T + 3T^{2} \)
5 \( 1 - 1.63iT - 5T^{2} \)
7 \( 1 + 0.615iT - 7T^{2} \)
11 \( 1 - 5.26iT - 11T^{2} \)
19 \( 1 + 1.59iT - 19T^{2} \)
23 \( 1 + 6.69T + 23T^{2} \)
29 \( 1 - 8.70T + 29T^{2} \)
31 \( 1 - 9.25iT - 31T^{2} \)
37 \( 1 - 1.63iT - 37T^{2} \)
41 \( 1 + 3.09iT - 41T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 2.31iT - 59T^{2} \)
61 \( 1 - 1.30T + 61T^{2} \)
67 \( 1 + 6.87iT - 67T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + 2.86iT - 73T^{2} \)
79 \( 1 + 1.71T + 79T^{2} \)
83 \( 1 - 5.58iT - 83T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + 2.28iT - 97T^{2} \)
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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

−12.29136711926108592581711591144, −11.93114968420777943015253873452, −10.61185146548672863823087910175, −10.12718308088741769098971767090, −8.434404272831652552000652758241, −7.14363346187577068268133672430, −6.83869525846039820661298663505, −5.61033120361403293257015803450, −4.70001463234104775303134069329, −2.53236142013933011042862029013, 0.76254463889581335996807597989, 2.66383027614377192116621346860, 4.23910883155658231231510505706, 5.61518410853617035456063900188, 6.40276978727871937593447791051, 8.007007680878284298036738361163, 9.166353591354152707194621044551, 10.25809872049499292865053476569, 11.08712563608160099747396909760, 11.96908809578194731189052112144

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