Properties

Label 2-1260-140.83-c0-0-3
Degree $2$
Conductor $1260$
Sign $0.229 + 0.973i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 5-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)10-s − 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 − i)17-s + 1.41i·19-s + 1.00i·20-s + (−1.00 + 1.00i)22-s + 25-s + (0.707 + 0.707i)28-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 5-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)10-s − 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 − i)17-s + 1.41i·19-s + 1.00i·20-s + (−1.00 + 1.00i)22-s + 25-s + (0.707 + 0.707i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9291878800\)
\(L(\frac12)\) \(\approx\) \(0.9291878800\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.722403030498742022998025751770, −9.033484104036436828095170000236, −8.263193592620122600055619196075, −7.49560741251107824503204334781, −6.49837795726161038754314142375, −5.52083872024425997673813104689, −4.38529663055047805334252454283, −3.34491376570338424905121438394, −2.22855267312832248037897920716, −1.12183159499626468381880677725, 1.75586070676143837343040325415, 2.33040060687394007776341711173, 4.49499505462255821884902086408, 5.13729643570304754357811602424, 6.01683665928884551687878272564, 6.82805233357577585111418683528, 7.54448674631309993284748732415, 8.693369829584176818558533126611, 9.075130608323551653660838450450, 9.832926153162524570280014628211

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