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Properties

Label 2-627-11.3-c1-0-23
Degree $2$
Conductor $627$
Sign $0.999 + 0.00420i$
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  + (1.74 − 1.26i)2-s + (0.309 + 0.951i)3-s + (0.815 − 2.51i)4-s + (2.24 + 1.63i)5-s + (1.74 + 1.26i)6-s + (−0.940 + 2.89i)7-s + (−0.425 − 1.30i)8-s + (−0.809 + 0.587i)9-s + 5.98·10-s + (0.962 − 3.17i)11-s + 2.63·12-s + (−2.31 + 1.67i)13-s + (2.02 + 6.23i)14-s + (−0.858 + 2.64i)15-s + (1.87 + 1.35i)16-s + (−2.15 − 1.56i)17-s + ⋯
L(s)  = 1  + (1.23 − 0.895i)2-s + (0.178 + 0.549i)3-s + (0.407 − 1.25i)4-s + (1.00 + 0.730i)5-s + (0.711 + 0.516i)6-s + (−0.355 + 1.09i)7-s + (−0.150 − 0.462i)8-s + (−0.269 + 0.195i)9-s + 1.89·10-s + (0.290 − 0.956i)11-s + 0.761·12-s + (−0.641 + 0.465i)13-s + (0.541 + 1.66i)14-s + (−0.221 + 0.682i)15-s + (0.467 + 0.339i)16-s + (−0.523 − 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(627\)    =    \(3 \cdot 11 \cdot 19\)
Sign: $0.999 + 0.00420i$
Analytic conductor: \(5.00662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{627} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 627,\ (\ :1/2),\ 0.999 + 0.00420i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.28892 - 0.00691895i\)
\(L(\frac12)\) \(\approx\) \(3.28892 - 0.00691895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.962 + 3.17i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-1.74 + 1.26i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.24 - 1.63i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.940 - 2.89i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.31 - 1.67i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.15 + 1.56i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 2.93T + 23T^{2} \)
29 \( 1 + (-2.73 + 8.42i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.39 + 6.09i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.201 - 0.621i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.52 + 7.78i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (-0.0728 - 0.224i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.93 - 1.40i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.795 + 2.44i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.22 + 5.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.05T + 67T^{2} \)
71 \( 1 + (3.74 + 2.72i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.02 - 3.16i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.83 - 6.42i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-14.5 - 10.5i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.08T + 89T^{2} \)
97 \( 1 + (-0.559 + 0.406i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.69362867878529367308180425293, −9.936352117185017180098338115916, −9.266462210257740234085157046241, −8.148452662356928908024549773359, −6.33367904138413958422803504627, −5.97026841821169584695860226403, −4.94200732217966189916115490734, −3.85352617606113868172911663670, −2.67176620687367197479850348063, −2.28869101492042807502503311655, 1.43037219315938329934581910749, 3.07188823499603041648821718627, 4.43392082079960876145752504282, 5.02788637576789611220590079419, 6.17010485295901101487300800194, 6.83710805283468452390938076365, 7.52273873835406857029024069148, 8.651848115478939008919826406440, 9.808252545897061492540757248343, 10.38589828303980747014883459873

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