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Properties

Label 2-63-63.5-c1-0-1
Degree $2$
Conductor $63$
Sign $-0.235 - 0.971i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
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  + 1.73i·2-s + 1.73i·3-s − 0.999·4-s + (−1.5 − 2.59i)5-s − 2.99·6-s + (2 − 1.73i)7-s + 1.73i·8-s − 2.99·9-s + (4.5 − 2.59i)10-s + (1.5 + 0.866i)11-s − 1.73i·12-s + (1.5 + 0.866i)13-s + (2.99 + 3.46i)14-s + (4.5 − 2.59i)15-s − 5·16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + 1.22i·2-s + 0.999i·3-s − 0.499·4-s + (−0.670 − 1.16i)5-s − 1.22·6-s + (0.755 − 0.654i)7-s + 0.612i·8-s − 0.999·9-s + (1.42 − 0.821i)10-s + (0.452 + 0.261i)11-s − 0.499i·12-s + (0.416 + 0.240i)13-s + (0.801 + 0.925i)14-s + (1.16 − 0.670i)15-s − 1.25·16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546085 + 0.694339i\)
\(L(\frac12)\) \(\approx\) \(0.546085 + 0.694339i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-7.5 + 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.35751208257009218854921448646, −14.70834666301896751611776578217, −13.49058812877495325330719049175, −11.77754183452763774526465769412, −10.83313545070113208300751799377, −8.959772757758035218124934610671, −8.380777143027558262929981240096, −6.91967742670604685606408541649, −5.09613453097170482446945321184, −4.34580941512952108439354401796, 2.07546201997506410131373966669, 3.57904951207528452812854647994, 6.20075781517012772590908988763, 7.49096196586189799249555304195, 8.824342011500182348229732239823, 10.77707182011915899439555582822, 11.28993963185115458104612883538, 12.16359412649119906651873937512, 13.21466091278928643433000240031, 14.56036387730836949818660069239

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