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Properties

Label 2-63-7.3-c2-0-3
Degree $2$
Conductor $63$
Sign $0.857 + 0.514i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)4-s + (6.5 − 2.59i)7-s + 12.1i·13-s + (−7.99 − 13.8i)16-s + (−31.5 + 18.1i)19-s + (−12.5 + 21.6i)25-s + (4 − 27.7i)28-s + (52.5 + 30.3i)31-s + (−36.5 − 63.2i)37-s + 61·43-s + (35.5 − 33.7i)49-s + (42 + 24.2i)52-s + (−84 + 48.4i)61-s − 63.9·64-s + (6.5 − 11.2i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (0.928 − 0.371i)7-s + 0.932i·13-s + (−0.499 − 0.866i)16-s + (−1.65 + 0.957i)19-s + (−0.5 + 0.866i)25-s + (0.142 − 0.989i)28-s + (1.69 + 0.977i)31-s + (−0.986 − 1.70i)37-s + 1.41·43-s + (0.724 − 0.689i)49-s + (0.807 + 0.466i)52-s + (−1.37 + 0.795i)61-s − 0.999·64-s + (0.0970 − 0.168i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.857 + 0.514i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30642 - 0.361701i\)
\(L(\frac12)\) \(\approx\) \(1.30642 - 0.361701i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-6.5 + 2.59i)T \)
good2 \( 1 + (-2 + 3.46i)T^{2} \)
5 \( 1 + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 12.1iT - 169T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (31.5 - 18.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + (-52.5 - 30.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (36.5 + 63.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 61T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (84 - 48.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + (94.5 + 54.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 193. iT - 9.40e3T^{2} \)
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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

−14.55427705800204850411987557067, −13.88619636377177409366953890782, −12.20905977802725208567214610228, −11.09115026222997729412476934948, −10.28408700440128297918597752403, −8.802465309030291298696486030989, −7.29863918295464248640678875541, −5.97953310096072569385332632094, −4.44364711489392849381763759020, −1.80347681501906246688704407860, 2.53876036153271695625248551063, 4.49539743087768569403012706023, 6.32906459864325633593013236774, 7.85482144546790696610603288259, 8.623334246964728123469544217439, 10.48528122824025194306920378730, 11.53869866258435274751122380070, 12.48044679524425182942558960747, 13.59028325535913707785545531298, 15.07902367657215131522183328558

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