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Properties

Label 2-61-61.13-c1-0-3
Degree $2$
Conductor $61$
Sign $0.326 + 0.945i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  + (1.03 − 1.78i)2-s − 0.326·3-s + (−1.12 − 1.94i)4-s + (−1.04 + 1.81i)5-s + (−0.336 + 0.583i)6-s + (−0.242 + 0.420i)7-s − 0.514·8-s − 2.89·9-s + (2.15 + 3.73i)10-s + 1.76·11-s + (0.367 + 0.636i)12-s + (−0.194 + 0.336i)13-s + (0.5 + 0.866i)14-s + (0.341 − 0.591i)15-s + (1.71 − 2.97i)16-s + (−2.78 − 4.81i)17-s + ⋯
L(s)  = 1  + (0.728 − 1.26i)2-s − 0.188·3-s + (−0.562 − 0.974i)4-s + (−0.467 + 0.809i)5-s + (−0.137 + 0.238i)6-s + (−0.0916 + 0.158i)7-s − 0.182·8-s − 0.964·9-s + (0.681 + 1.18i)10-s + 0.532·11-s + (0.106 + 0.183i)12-s + (−0.0538 + 0.0932i)13-s + (0.133 + 0.231i)14-s + (0.0881 − 0.152i)15-s + (0.429 − 0.744i)16-s + (−0.674 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.326 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.876468 - 0.624313i\)
\(L(\frac12)\) \(\approx\) \(0.876468 - 0.624313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (7.32 - 2.70i)T \)
good2 \( 1 + (-1.03 + 1.78i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.326T + 3T^{2} \)
5 \( 1 + (1.04 - 1.81i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.242 - 0.420i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.76T + 11T^{2} \)
13 \( 1 + (0.194 - 0.336i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.78 + 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.86 - 4.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.68T + 23T^{2} \)
29 \( 1 + (4.99 + 8.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.82 - 3.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.33T + 37T^{2} \)
41 \( 1 - 3.40T + 41T^{2} \)
43 \( 1 + (-5.13 + 8.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.62 - 9.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.580T + 53T^{2} \)
59 \( 1 + (2.75 - 4.78i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (-3.47 + 6.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.36 - 9.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.656 + 1.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.30 - 5.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.34 + 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (3.40 + 5.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.32868218080562524901919053017, −13.81376107840626597770561140183, −12.19514698755947737763171354285, −11.61237018121274004933427659830, −10.76892630327112056850695003106, −9.441388790115660374229376653564, −7.57598609191210149074269317139, −5.82240105095123565002523037859, −4.03433505687466146751887119921, −2.68290853282709917790083162647, 4.08276088685714815715612202818, 5.36788629326955566729705904168, 6.54910710046626973401458732000, 7.956907821288219674438175857677, 8.975836301789257050008708089493, 10.96536925327616321034634748226, 12.26397713188152791484823542727, 13.31653443061138282252954366945, 14.34903179203334228356368177216, 15.29469710552776025613291513130

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