Properties

Label 2-630-9.4-c1-0-11
Degree $2$
Conductor $630$
Sign $0.993 - 0.112i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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  + (0.5 + 0.866i)2-s + (−1.18 − 1.26i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 − 1.65i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−0.186 + 2.99i)9-s + 0.999·10-s + (0.5 + 0.866i)11-s + (1.68 − 0.396i)12-s + (1.68 − 2.92i)13-s + (−0.499 + 0.866i)14-s + (−1.68 + 0.396i)15-s + (−0.5 − 0.866i)16-s + 3·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.684 − 0.728i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.204 − 0.677i)6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (−0.0620 + 0.998i)9-s + 0.316·10-s + (0.150 + 0.261i)11-s + (0.486 − 0.114i)12-s + (0.467 − 0.809i)13-s + (−0.133 + 0.231i)14-s + (−0.435 + 0.102i)15-s + (−0.125 − 0.216i)16-s + 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.993 - 0.112i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52578 + 0.0858746i\)
\(L(\frac12)\) \(\approx\) \(1.52578 + 0.0858746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.18 + 1.26i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 + 2.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 4.37T + 19T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + (-3.18 + 5.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.81 + 3.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.313 - 0.543i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.744T + 53T^{2} \)
59 \( 1 + (5.55 - 9.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.74 - 9.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.18 - 3.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.1T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + (-0.0584 - 0.101i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.313 + 0.543i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + (8.24 + 14.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.73075247117028466572718911488, −9.702666300541608913025768018942, −8.586242957513977488577113961227, −7.79408664895834039529608396026, −7.03800976091988956700420274960, −5.84466911328481910318102402959, −5.50617384168350163739893976295, −4.37282720557237320242181858747, −2.78139933102383953769933984022, −1.09226687374697451825228247971, 1.20318110210742165964377202696, 3.03553376660405416946557292422, 3.97006636284409949837160487469, 4.93175637271345509682462195302, 5.89740333663092181857702154342, 6.69530989121447698994434146272, 8.004057929936487607271834849397, 9.406847884892188608424330952853, 9.735167055619096912181309567608, 10.78885710581712687059998160994

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