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Properties

Label 2-630-5.2-c2-0-24
Degree $2$
Conductor $630$
Sign $0.574 + 0.818i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + 2i·4-s + (−0.294 − 4.99i)5-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s + (4.69 − 5.28i)10-s + 3.32·11-s + (−7.80 + 7.80i)13-s − 3.74i·14-s − 4·16-s + (13.6 + 13.6i)17-s − 37.1i·19-s + (9.98 − 0.588i)20-s + (3.32 + 3.32i)22-s + (31.5 − 31.5i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.0588 − 0.998i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.469 − 0.528i)10-s + 0.302·11-s + (−0.600 + 0.600i)13-s − 0.267i·14-s − 0.250·16-s + (0.803 + 0.803i)17-s − 1.95i·19-s + (0.499 − 0.0294i)20-s + (0.151 + 0.151i)22-s + (1.36 − 1.36i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.574 + 0.818i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.901625416\)
\(L(\frac12)\) \(\approx\) \(1.901625416\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (0.294 + 4.99i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 - 3.32T + 121T^{2} \)
13 \( 1 + (7.80 - 7.80i)T - 169iT^{2} \)
17 \( 1 + (-13.6 - 13.6i)T + 289iT^{2} \)
19 \( 1 + 37.1iT - 361T^{2} \)
23 \( 1 + (-31.5 + 31.5i)T - 529iT^{2} \)
29 \( 1 + 55.3iT - 841T^{2} \)
31 \( 1 + 4.54T + 961T^{2} \)
37 \( 1 + (46.8 + 46.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 9.94T + 1.68e3T^{2} \)
43 \( 1 + (-23.0 + 23.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-6.38 - 6.38i)T + 2.20e3iT^{2} \)
53 \( 1 + (-42.9 + 42.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 59.2iT - 3.48e3T^{2} \)
61 \( 1 - 47.1T + 3.72e3T^{2} \)
67 \( 1 + (-8.48 - 8.48i)T + 4.48e3iT^{2} \)
71 \( 1 + 85.6T + 5.04e3T^{2} \)
73 \( 1 + (34.7 - 34.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 96.4iT - 6.24e3T^{2} \)
83 \( 1 + (19.6 - 19.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 43.0iT - 7.92e3T^{2} \)
97 \( 1 + (-88.5 - 88.5i)T + 9.40e3iT^{2} \)
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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

−10.17423686997048072192994814206, −9.106885707166594899832160154095, −8.620690102983195620940407643247, −7.42858270248283129482565804281, −6.71922106437911050315048008957, −5.59106960176844000978933649770, −4.68842396666051556679534207036, −3.93957033526819700208184352783, −2.43012689429367074803384058884, −0.61002720088285698509076690091, 1.52558412991224866717867271577, 3.08092008195078530397217855969, 3.46324298439139721821729228085, 5.08655177338419962954324191480, 5.82581445739466591680837096637, 6.96379710986845287412654745225, 7.65947477266496334171730271234, 9.005049749590961264244145404729, 9.999351395426471410709133139131, 10.43791439837879818584070552236

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