Properties

Label 2-126-63.5-c1-0-1
Degree $2$
Conductor $126$
Sign $0.835 - 0.550i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
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 + (−0.890 + 1.48i)3-s − 4-s + (1.14 + 1.97i)5-s + (1.48 + 0.890i)6-s + (1.42 + 2.23i)7-s + i·8-s + (−1.41 − 2.64i)9-s + (1.97 − 1.14i)10-s + (−0.946 − 0.546i)11-s + (0.890 − 1.48i)12-s + (5.91 + 3.41i)13-s + (2.23 − 1.42i)14-s + (−3.95 − 0.0659i)15-s + 16-s + (−3.35 − 5.81i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.514 + 0.857i)3-s − 0.5·4-s + (0.510 + 0.883i)5-s + (0.606 + 0.363i)6-s + (0.537 + 0.842i)7-s + 0.353i·8-s + (−0.470 − 0.882i)9-s + (0.624 − 0.360i)10-s + (−0.285 − 0.164i)11-s + (0.257 − 0.428i)12-s + (1.64 + 0.947i)13-s + (0.596 − 0.380i)14-s + (−1.02 − 0.0170i)15-s + 0.250·16-s + (−0.814 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.835 - 0.550i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.835 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.926522 + 0.277736i\)
\(L(\frac12)\) \(\approx\) \(0.926522 + 0.277736i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.890 - 1.48i)T \)
7 \( 1 + (-1.42 - 2.23i)T \)
good5 \( 1 + (-1.14 - 1.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.946 + 0.546i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.91 - 3.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.35 + 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.47 + 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.38 - 1.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.59 + 0.923i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.02iT - 31T^{2} \)
37 \( 1 + (-3.57 + 6.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.45 + 4.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.74 + 6.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.80T + 47T^{2} \)
53 \( 1 + (-0.222 + 0.128i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.94T + 59T^{2} \)
61 \( 1 - 1.33iT - 61T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + 0.233iT - 71T^{2} \)
73 \( 1 + (-5.89 + 3.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.27T + 79T^{2} \)
83 \( 1 + (2.91 + 5.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.99 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.13 - 2.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.61604431900058545991096620564, −12.03209485539709092730612030486, −11.19584179904763282270326623324, −10.69877545056275802345410360359, −9.403464748239561217391805347728, −8.669746322712515048480156644571, −6.56636877959474855618117456922, −5.47939295210940538335708743557, −4.07792756132106153823257734500, −2.48241903990028293608058219074, 1.33289979862242666992768682035, 4.34428361106753044321314579809, 5.69011459079965489777860593833, 6.50391673707951521804243901369, 8.067473647195270617687327449826, 8.455428803867469118445867042379, 10.28927291094765305729893036961, 11.17461374748683766714677680624, 12.86941844469528458343671913798, 13.07690848733152196642238407448

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