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Properties

Label 2-1407-1407.1376-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.998 + 0.0576i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
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  + (0.981 − 0.189i)3-s + (−0.959 − 0.281i)4-s + (0.580 + 0.814i)7-s + (0.928 − 0.371i)9-s + (−0.995 − 0.0950i)12-s + (0.283 + 0.0270i)13-s + (0.841 + 0.540i)16-s + (0.0930 + 0.647i)19-s + (0.723 + 0.690i)21-s + (−0.995 − 0.0950i)25-s + (0.841 − 0.540i)27-s + (−0.327 − 0.945i)28-s + (0.698 − 1.53i)31-s + (−0.995 + 0.0950i)36-s + 1.16·37-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)3-s + (−0.959 − 0.281i)4-s + (0.580 + 0.814i)7-s + (0.928 − 0.371i)9-s + (−0.995 − 0.0950i)12-s + (0.283 + 0.0270i)13-s + (0.841 + 0.540i)16-s + (0.0930 + 0.647i)19-s + (0.723 + 0.690i)21-s + (−0.995 − 0.0950i)25-s + (0.841 − 0.540i)27-s + (−0.327 − 0.945i)28-s + (0.698 − 1.53i)31-s + (−0.995 + 0.0950i)36-s + 1.16·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.998 + 0.0576i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.998 + 0.0576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.345595601\)
\(L(\frac12)\) \(\approx\) \(1.345595601\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (-0.580 - 0.814i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
good2 \( 1 + (0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.995 + 0.0950i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.283 - 0.0270i)T + (0.981 + 0.189i)T^{2} \)
17 \( 1 + (-0.0475 - 0.998i)T^{2} \)
19 \( 1 + (-0.0930 - 0.647i)T + (-0.959 + 0.281i)T^{2} \)
23 \( 1 + (0.142 - 0.989i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 - 1.16T + T^{2} \)
41 \( 1 + (-0.0475 - 0.998i)T^{2} \)
43 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.928 + 0.371i)T^{2} \)
53 \( 1 + (-0.0475 + 0.998i)T^{2} \)
59 \( 1 + (0.327 - 0.945i)T^{2} \)
61 \( 1 + (0.550 - 0.353i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.888 - 0.458i)T^{2} \)
73 \( 1 + (0.0845 + 0.0436i)T + (0.580 + 0.814i)T^{2} \)
79 \( 1 + (1.15 + 0.110i)T + (0.981 + 0.189i)T^{2} \)
83 \( 1 + (-0.580 + 0.814i)T^{2} \)
89 \( 1 + (-0.928 - 0.371i)T^{2} \)
97 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.566376618613340228411248475676, −8.964034967958319965989815722890, −8.059370908603972490339187614691, −7.86428628449545752819360236698, −6.36111443093657182774458027741, −5.58603387108554621787987659377, −4.51849321839621010940689262974, −3.80864567770204940712699119853, −2.58425549109316993553227224058, −1.46301067983109296508966796487, 1.35482689751727253217668834437, 2.87460168626675332518954243346, 3.83970903186301142543150746935, 4.47526579959159103066637815620, 5.28482804260984219779908239562, 6.75267124829451285667000979371, 7.62777026907328779714011826664, 8.231178254793012061930142095984, 8.852678032190442867432590348884, 9.697997640726645645775980489537

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