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Properties

Label 2-627-19.7-c1-0-20
Degree $2$
Conductor $627$
Sign $-0.853 - 0.521i$
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  + (−1.35 − 2.35i)2-s + (−0.5 − 0.866i)3-s + (−2.69 + 4.65i)4-s + (−0.467 − 0.809i)5-s + (−1.35 + 2.35i)6-s + 4.17·7-s + 9.18·8-s + (−0.499 + 0.866i)9-s + (−1.27 + 2.20i)10-s − 11-s + 5.38·12-s + (−0.486 + 0.841i)13-s + (−5.67 − 9.83i)14-s + (−0.467 + 0.809i)15-s + (−7.09 − 12.2i)16-s + (−2.03 − 3.51i)17-s + ⋯
L(s)  = 1  + (−0.960 − 1.66i)2-s + (−0.288 − 0.499i)3-s + (−1.34 + 2.32i)4-s + (−0.209 − 0.362i)5-s + (−0.554 + 0.960i)6-s + 1.57·7-s + 3.24·8-s + (−0.166 + 0.288i)9-s + (−0.401 + 0.695i)10-s − 0.301·11-s + 1.55·12-s + (−0.134 + 0.233i)13-s + (−1.51 − 2.62i)14-s + (−0.120 + 0.209i)15-s + (−1.77 − 3.07i)16-s + (−0.492 − 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(627\)    =    \(3 \cdot 11 \cdot 19\)
Sign: $-0.853 - 0.521i$
Analytic conductor: \(5.00662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{627} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 627,\ (\ :1/2),\ -0.853 - 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.173483 + 0.616406i\)
\(L(\frac12)\) \(\approx\) \(0.173483 + 0.616406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.11 + 1.43i)T \)
good2 \( 1 + (1.35 + 2.35i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.467 + 0.809i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.17T + 7T^{2} \)
13 \( 1 + (0.486 - 0.841i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.03 + 3.51i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.65 + 8.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0766 - 0.132i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + 4.81T + 37T^{2} \)
41 \( 1 + (2.91 + 5.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.79 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.867 - 1.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.367 - 0.637i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.98 + 5.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.37 + 4.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.59 + 6.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.90 - 6.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.61 + 9.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.31 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.35T + 83T^{2} \)
89 \( 1 + (7.58 - 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.525 + 0.910i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49226753042344901985530161637, −9.139428848236019520256532048795, −8.542302857346087208568082662199, −7.926086490271844637665717217509, −6.92392865008310109874856724721, −4.87328076062000119498681078889, −4.43896090829304772892719411191, −2.69839233445214445121569845715, −1.81533541244290032747277457749, −0.53892193752777882306174372226, 1.54721344136695604898172448670, 4.10161877799114139805574222748, 5.16312878727932510584672043101, 5.65714728751514833247041291262, 6.92762193503113121110238844617, 7.58298377916991779931009272764, 8.453985600255979263235490814840, 8.983100943489349499560726565239, 10.15467158617182563131755966657, 10.81115314071344968414474544047

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