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Properties

Label 2-1412-1412.295-c0-0-0
Degree $2$
Conductor $1412$
Sign $-0.958 - 0.285i$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (−0.557 − 1.89i)5-s + (−0.654 − 0.755i)8-s + (−0.841 + 0.540i)9-s + 1.97i·10-s + (−0.512 + 0.234i)13-s + (0.415 + 0.909i)16-s + (−1.10 − 1.27i)17-s + (0.959 − 0.281i)18-s + (0.557 − 1.89i)20-s + (−2.45 + 1.57i)25-s + (0.557 − 0.0801i)26-s + (−0.698 + 0.449i)29-s + (−0.142 − 0.989i)32-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (−0.557 − 1.89i)5-s + (−0.654 − 0.755i)8-s + (−0.841 + 0.540i)9-s + 1.97i·10-s + (−0.512 + 0.234i)13-s + (0.415 + 0.909i)16-s + (−1.10 − 1.27i)17-s + (0.959 − 0.281i)18-s + (0.557 − 1.89i)20-s + (−2.45 + 1.57i)25-s + (0.557 − 0.0801i)26-s + (−0.698 + 0.449i)29-s + (−0.142 − 0.989i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.958 - 0.285i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ -0.958 - 0.285i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2220019500\)
\(L(\frac12)\) \(\approx\) \(0.2220019500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
353 \( 1 + (0.654 - 0.755i)T \)
good3 \( 1 + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.557 + 1.89i)T + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.512 - 0.234i)T + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
23 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (-1.14 + 0.989i)T + (0.142 - 0.989i)T^{2} \)
41 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.415 - 0.909i)T^{2} \)
53 \( 1 + (-0.304 - 1.03i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (1.68 + 1.08i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (1.49 + 1.29i)T + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
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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

−9.141263597509693457765914678114, −8.711306223063145582847570146663, −7.88429911157192163555275599486, −7.32460313838643717389686481587, −5.99667604744983909657174630258, −4.99500786140672571372383253087, −4.28777645479726185514169998168, −2.87690889034088062154611746121, −1.69468186199342591748214176217, −0.23183492949102916020353870653, 2.20307845115481511138341663527, 2.97946513411571027940822607050, 3.97946006200580644007264198669, 5.69751605081614689769931192825, 6.42486607521161957455073533226, 6.92581437061080215154929455840, 7.82300276439750225518190644634, 8.418849422641007839205049457204, 9.431714140775694626480349394261, 10.26298539395883408656072366610

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