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Properties

Label 2-63-63.59-c1-0-1
Degree $2$
Conductor $63$
Sign $0.615 - 0.788i$
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  + (1.30 + 0.755i)2-s + (−0.919 + 1.46i)3-s + (0.140 + 0.242i)4-s − 0.775·5-s + (−2.31 + 1.22i)6-s + (2.05 − 1.66i)7-s − 2.59i·8-s + (−1.30 − 2.69i)9-s + (−1.01 − 0.585i)10-s + 3.84i·11-s + (−0.485 − 0.0176i)12-s + (−2.54 − 1.46i)13-s + (3.94 − 0.618i)14-s + (0.713 − 1.13i)15-s + (2.24 − 3.88i)16-s + (−2.69 + 4.67i)17-s + ⋯
L(s)  = 1  + (0.924 + 0.533i)2-s + (−0.531 + 0.847i)3-s + (0.0700 + 0.121i)4-s − 0.346·5-s + (−0.943 + 0.500i)6-s + (0.778 − 0.628i)7-s − 0.918i·8-s + (−0.435 − 0.899i)9-s + (−0.320 − 0.185i)10-s + 1.15i·11-s + (−0.140 − 0.00508i)12-s + (−0.705 − 0.407i)13-s + (1.05 − 0.165i)14-s + (0.184 − 0.294i)15-s + (0.560 − 0.970i)16-s + (−0.654 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995630 + 0.485754i\)
\(L(\frac12)\) \(\approx\) \(0.995630 + 0.485754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.919 - 1.46i)T \)
7 \( 1 + (-2.05 + 1.66i)T \)
good2 \( 1 + (-1.30 - 0.755i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.775T + 5T^{2} \)
11 \( 1 - 3.84iT - 11T^{2} \)
13 \( 1 + (2.54 + 1.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.69 - 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.376 - 0.217i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.0557iT - 23T^{2} \)
29 \( 1 + (0.187 - 0.108i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.67 + 3.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.14 - 5.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.78 + 6.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.42 - 11.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.482 + 0.836i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.46 + 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.56 + 2.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.01 + 1.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 3.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (7.05 + 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.48 + 4.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.31 + 7.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.82 - 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.24 + 0.716i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.06106010155907892675690513361, −14.45509413221974247264947816898, −13.02712239835909865073719020792, −11.92016212693724008709557438222, −10.60185021785359936784217272372, −9.659119248259383324674194324177, −7.72997844310302087134852417632, −6.27578699173217066747188596408, −4.81569658156642125329987650892, −4.12982533312461995788384839088, 2.53390167019242355721496667255, 4.66078822304143354051068806856, 5.84209608895373911285051621671, 7.60455067945044150213934532822, 8.748221862310847113377947567043, 11.09663561283697067421962711733, 11.65123059625816529896464584638, 12.44433690431247595527148329578, 13.67739878237072159575100534692, 14.27471062154701619377856573489

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