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Properties

Label 2-627-19.11-c1-0-0
Degree $2$
Conductor $627$
Sign $0.327 + 0.944i$
Analytic cond. $5.00662$
Root an. cond. $2.23754$
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  + (−0.794 + 1.37i)2-s + (−0.5 + 0.866i)3-s + (−0.263 − 0.455i)4-s + (−1.98 + 3.44i)5-s + (−0.794 − 1.37i)6-s + 0.0905·7-s − 2.34·8-s + (−0.499 − 0.866i)9-s + (−3.16 − 5.47i)10-s − 11-s + 0.526·12-s + (−1.02 − 1.77i)13-s + (−0.0719 + 0.124i)14-s + (−1.98 − 3.44i)15-s + (2.38 − 4.13i)16-s + (−0.207 + 0.359i)17-s + ⋯
L(s)  = 1  + (−0.561 + 0.973i)2-s + (−0.288 + 0.499i)3-s + (−0.131 − 0.227i)4-s + (−0.889 + 1.54i)5-s + (−0.324 − 0.561i)6-s + 0.0342·7-s − 0.828·8-s + (−0.166 − 0.288i)9-s + (−0.999 − 1.73i)10-s − 0.301·11-s + 0.151·12-s + (−0.284 − 0.492i)13-s + (−0.0192 + 0.0333i)14-s + (−0.513 − 0.889i)15-s + (0.596 − 1.03i)16-s + (−0.0503 + 0.0871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(627\)    =    \(3 \cdot 11 \cdot 19\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(5.00662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{627} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 627,\ (\ :1/2),\ 0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.265939 - 0.189340i\)
\(L(\frac12)\) \(\approx\) \(0.265939 - 0.189340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
19 \( 1 + (-0.524 - 4.32i)T \)
good2 \( 1 + (0.794 - 1.37i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.98 - 3.44i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.0905T + 7T^{2} \)
13 \( 1 + (1.02 + 1.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.207 - 0.359i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.0351 - 0.0608i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.91 - 6.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.40T + 31T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 + (2.54 - 4.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.37 - 5.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.79 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.38 + 4.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.08 + 8.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.95 + 3.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.74 + 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.00 - 6.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.00 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.92 - 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.16T + 83T^{2} \)
89 \( 1 + (-1.38 - 2.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.63 - 2.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19748465375654414603000285676, −10.30462132776299355806407666782, −9.667134692763502848933375543061, −8.217471776137255609839056461355, −7.895892962115988027978585964212, −6.82173873337306118480819631661, −6.33714873002660654737586355796, −5.10366789626396301021249361276, −3.61943796346771066772990824498, −2.89293342460774757850786765877, 0.23782174260916193640025221720, 1.30690624874635620244968035217, 2.66702108469781293803875890824, 4.21974985222137445343471476183, 5.09431234602299442916850083015, 6.26765202523811461502811187931, 7.54228176580241818279209928847, 8.375312145286623064450972841954, 9.084670340499075947888794474646, 9.851699788759877184399724039351

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