Properties

Label 2-630-5.4-c1-0-4
Degree $2$
Conductor $630$
Sign $-0.447 - 0.894i$
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  + i·2-s − 4-s + (1 + 2i)5-s i·7-s i·8-s + (−2 + i)10-s + 2·11-s + 2i·13-s + 14-s + 16-s + 8i·17-s + 2·19-s + (−1 − 2i)20-s + 2i·22-s + (−3 + 4i)25-s − 2·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.447 + 0.894i)5-s − 0.377i·7-s − 0.353i·8-s + (−0.632 + 0.316i)10-s + 0.603·11-s + 0.554i·13-s + 0.267·14-s + 0.250·16-s + 1.94i·17-s + 0.458·19-s + (−0.223 − 0.447i)20-s + 0.426i·22-s + (−0.600 + 0.800i)25-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765249 + 1.23819i\)
\(L(\frac12)\) \(\approx\) \(0.765249 + 1.23819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-1 - 2i)T \)
7 \( 1 + iT \)
good11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.74423386917910053261749379624, −9.949401906157975379336025401173, −9.146495760488311143127034002750, −8.107620373490479955672406980675, −7.21119637536265728910856406109, −6.39715154140196961986798094406, −5.77585435797111961215641278645, −4.33070952328280847170156845796, −3.42869511443778481440939090764, −1.74385858215399389597469014219, 0.854284030856195579160614206449, 2.26787165417138267047105949320, 3.49040150299360966035809250033, 4.86017933399314605250365526660, 5.38839019017613361890560745660, 6.67291692828615502546291759996, 7.926107802115565613681613333342, 8.841540788479496506418318143254, 9.520520359786936001837909184030, 10.11390651499434925418445489318

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