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Properties

Label 2-1407-1407.851-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.947 - 0.320i$
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  + (−0.786 − 0.618i)3-s + (0.841 + 0.540i)4-s + (0.981 − 0.189i)7-s + (0.235 + 0.971i)9-s + (−0.327 − 0.945i)12-s + (0.627 + 1.81i)13-s + (0.415 + 0.909i)16-s + (−1.78 + 0.523i)19-s + (−0.888 − 0.458i)21-s + (−0.327 − 0.945i)25-s + (0.415 − 0.909i)27-s + (0.928 + 0.371i)28-s + (−0.544 − 0.627i)31-s + (−0.327 + 0.945i)36-s + 1.96·37-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)3-s + (0.841 + 0.540i)4-s + (0.981 − 0.189i)7-s + (0.235 + 0.971i)9-s + (−0.327 − 0.945i)12-s + (0.627 + 1.81i)13-s + (0.415 + 0.909i)16-s + (−1.78 + 0.523i)19-s + (−0.888 − 0.458i)21-s + (−0.327 − 0.945i)25-s + (0.415 − 0.909i)27-s + (0.928 + 0.371i)28-s + (−0.544 − 0.627i)31-s + (−0.327 + 0.945i)36-s + 1.96·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128275751\)
\(L(\frac12)\) \(\approx\) \(1.128275751\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 + 0.618i)T \)
7 \( 1 + (-0.981 + 0.189i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
good2 \( 1 + (-0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.327 + 0.945i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (-0.627 - 1.81i)T + (-0.786 + 0.618i)T^{2} \)
17 \( 1 + (-0.580 - 0.814i)T^{2} \)
19 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 - 1.96T + T^{2} \)
41 \( 1 + (-0.580 - 0.814i)T^{2} \)
43 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.235 - 0.971i)T^{2} \)
53 \( 1 + (-0.580 + 0.814i)T^{2} \)
59 \( 1 + (-0.928 + 0.371i)T^{2} \)
61 \( 1 + (-0.771 + 1.68i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (0.995 + 0.0950i)T^{2} \)
73 \( 1 + (1.15 - 0.110i)T + (0.981 - 0.189i)T^{2} \)
79 \( 1 + (0.642 + 1.85i)T + (-0.786 + 0.618i)T^{2} \)
83 \( 1 + (-0.981 - 0.189i)T^{2} \)
89 \( 1 + (-0.235 + 0.971i)T^{2} \)
97 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.02323075089196613081929415458, −8.655712702651717384900909228393, −8.080152921866672700471633111379, −7.31135133217359432644148407907, −6.40082463348320959867045907423, −6.10611932482716519464987344913, −4.61156249027893474450732052315, −4.02845324137134111795621969030, −2.22602404404971745005613950623, −1.67669036850089124890798551909, 1.14031724427529026021248868780, 2.51999925036823750895001367835, 3.77209988245989589389276445565, 4.91945370468212599069846182425, 5.58105871401545082862346843623, 6.19321381135707460253749773954, 7.16427734558490007628209122450, 8.113988397635704044585583089902, 8.932413009003485072401664229139, 10.07852041952928291805120372670

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