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Properties

Label 2-63-63.5-c1-0-5
Degree $2$
Conductor $63$
Sign $-0.708 + 0.706i$
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  − 2.09i·2-s + (−1.72 − 0.109i)3-s − 2.39·4-s + (−1.04 − 1.80i)5-s + (−0.228 + 3.62i)6-s + (2.60 + 0.486i)7-s + 0.819i·8-s + (2.97 + 0.377i)9-s + (−3.79 + 2.18i)10-s + (2.79 + 1.61i)11-s + (4.13 + 0.260i)12-s + (−2.68 − 1.55i)13-s + (1.01 − 5.44i)14-s + (1.60 + 3.24i)15-s − 3.06·16-s + (0.816 + 1.41i)17-s + ⋯
L(s)  = 1  − 1.48i·2-s + (−0.998 − 0.0629i)3-s − 1.19·4-s + (−0.467 − 0.809i)5-s + (−0.0933 + 1.47i)6-s + (0.982 + 0.183i)7-s + 0.289i·8-s + (0.992 + 0.125i)9-s + (−1.19 + 0.692i)10-s + (0.843 + 0.486i)11-s + (1.19 + 0.0753i)12-s + (−0.745 − 0.430i)13-s + (0.272 − 1.45i)14-s + (0.415 + 0.837i)15-s − 0.766·16-s + (0.197 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.708 + 0.706i$
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: \((2,\ 63,\ (\ :1/2),\ -0.708 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263581 - 0.637531i\)
\(L(\frac12)\) \(\approx\) \(0.263581 - 0.637531i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.109i)T \)
7 \( 1 + (-2.60 - 0.486i)T \)
good2 \( 1 + 2.09iT - 2T^{2} \)
5 \( 1 + (1.04 + 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.79 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.68 + 1.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.816 - 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.79 - 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.00 - 0.580i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.05 - 4.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.96iT - 31T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.35 + 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.974 + 1.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.13T + 47T^{2} \)
53 \( 1 + (5.27 - 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.96T + 59T^{2} \)
61 \( 1 + 4.79iT - 61T^{2} \)
67 \( 1 + 0.673T + 67T^{2} \)
71 \( 1 - 7.01iT - 71T^{2} \)
73 \( 1 + (2.96 - 1.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + (1.54 + 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.07 - 1.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.33851666482971835225048790024, −12.65713254335315324830814577833, −12.19287227523993843810392051494, −11.43140117922253983916168152479, −10.36146572877607892469766782145, −9.160472024019308950446626910619, −7.46626864511964229505312639823, −5.27815577660125468656257582310, −4.12156770988544340531875106824, −1.43063714994403906802640384247, 4.45944500205841727810836742856, 5.75542004776216633202045767091, 7.01454383797415794628723401480, 7.70120533996672158585555452087, 9.457453522824093592144545745665, 11.23229756018065290450012918045, 11.67006077614705359205568083927, 13.67499310706359552579402066087, 14.68660496765702840777286985010, 15.36448057461452225913737069888

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