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Properties

Label 2-756-7.3-c0-0-0
Degree $2$
Conductor $756$
Sign $0.922 + 0.386i$
Analytic cond. $0.377293$
Root an. cond. $0.614241$
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  + (0.5 − 0.866i)7-s + (1.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (1 + 1.73i)37-s + 43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 − 0.866i)73-s + (−1 − 1.73i)79-s + 1.73i·97-s + (−0.5 + 0.866i)109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (1.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (1 + 1.73i)37-s + 43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 − 0.866i)73-s + (−1 − 1.73i)79-s + 1.73i·97-s + (−0.5 + 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(0.377293\)
Root analytic conductor: \(0.614241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :0),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017409161\)
\(L(\frac12)\) \(\approx\) \(1.017409161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−10.52763465050665019733674135406, −9.640117839765748790219414294167, −8.897696436817363988728595590791, −7.57121436215882822954182434071, −7.39122724278161595238615136385, −6.04622910470620979602235280797, −5.04931520994212375136737243929, −4.10737781036924716793337064531, −2.96844602323825526379510974099, −1.34901770569908910157720622206, 1.72959302221823731376959918275, 2.99628300238794761710597513424, 4.22198973540775874696625740554, 5.41675416550233053352327716090, 5.95869670364095433598890288696, 7.32121848030643215348758627346, 7.981470997200612890847591784955, 8.993744850455697861459664885527, 9.609291742336016477735603166610, 10.68102768420910412258500959013

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