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Properties

Label 2-56e2-448.219-c0-0-0
Degree $2$
Conductor $3136$
Sign $0.231 + 0.972i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.382 − 0.923i)8-s + (0.991 + 0.130i)9-s + (−0.172 + 0.349i)11-s + (−0.866 − 0.499i)16-s + (0.866 − 0.499i)18-s + (0.0761 + 0.382i)22-s + (1.83 + 0.241i)23-s + (−0.130 − 0.991i)25-s + (−0.324 − 0.216i)29-s + (−0.991 + 0.130i)32-s + (0.382 − 0.923i)36-s + (1.09 − 1.25i)37-s + (−0.923 + 0.617i)43-s + (0.293 + 0.257i)44-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.382 − 0.923i)8-s + (0.991 + 0.130i)9-s + (−0.172 + 0.349i)11-s + (−0.866 − 0.499i)16-s + (0.866 − 0.499i)18-s + (0.0761 + 0.382i)22-s + (1.83 + 0.241i)23-s + (−0.130 − 0.991i)25-s + (−0.324 − 0.216i)29-s + (−0.991 + 0.130i)32-s + (0.382 − 0.923i)36-s + (1.09 − 1.25i)37-s + (−0.923 + 0.617i)43-s + (0.293 + 0.257i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 0.231 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.091007386\)
\(L(\frac12)\) \(\approx\) \(2.091007386\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 \)
good3 \( 1 + (-0.991 - 0.130i)T^{2} \)
5 \( 1 + (0.130 + 0.991i)T^{2} \)
11 \( 1 + (0.172 - 0.349i)T + (-0.608 - 0.793i)T^{2} \)
13 \( 1 + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.793 + 0.608i)T^{2} \)
23 \( 1 + (-1.83 - 0.241i)T + (0.965 + 0.258i)T^{2} \)
29 \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.09 + 1.25i)T + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.996 + 0.491i)T + (0.608 + 0.793i)T^{2} \)
59 \( 1 + (-0.793 + 0.608i)T^{2} \)
61 \( 1 + (-0.608 + 0.793i)T^{2} \)
67 \( 1 + (0.108 - 1.65i)T + (-0.991 - 0.130i)T^{2} \)
71 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.258 - 0.965i)T^{2} \)
79 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + (-0.923 + 0.382i)T^{2} \)
89 \( 1 + (0.258 - 0.965i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.944211212391726531185199639043, −7.78670913485773292247061704434, −7.06665278529893768436281622479, −6.40237702059397680350376712210, −5.45443899697092054768217306907, −4.69409921003128594997978811311, −4.10801078888428028317514953904, −3.09832363605316833940035781323, −2.18841126606335364804433677258, −1.14246717468407129342417335677, 1.51052227791682814238490750253, 2.87949674883406063811695424499, 3.55065765431094275131039021356, 4.57783475361866083162360867172, 5.07998855463531483987227396294, 6.03196986842117318922916465413, 6.79288517541063635403471011117, 7.34455843561571446159332712134, 8.068250956787334137846205343342, 8.933158801917758394366259365184

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