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Properties

Label 2-1407-1407.389-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.918 - 0.395i$
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  + (−0.327 − 0.945i)3-s + (0.723 + 0.690i)4-s + (−0.142 + 0.989i)7-s + (−0.786 + 0.618i)9-s + (0.415 − 0.909i)12-s + (0.283 − 0.0270i)13-s + (0.0475 + 0.998i)16-s + (0.514 + 0.404i)19-s + (0.981 − 0.189i)21-s + (0.580 + 0.814i)25-s + (0.841 + 0.540i)27-s + (−0.786 + 0.618i)28-s + (0.975 − 1.37i)31-s + (−0.995 − 0.0950i)36-s + (−0.580 + 1.00i)37-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)3-s + (0.723 + 0.690i)4-s + (−0.142 + 0.989i)7-s + (−0.786 + 0.618i)9-s + (0.415 − 0.909i)12-s + (0.283 − 0.0270i)13-s + (0.0475 + 0.998i)16-s + (0.514 + 0.404i)19-s + (0.981 − 0.189i)21-s + (0.580 + 0.814i)25-s + (0.841 + 0.540i)27-s + (−0.786 + 0.618i)28-s + (0.975 − 1.37i)31-s + (−0.995 − 0.0950i)36-s + (−0.580 + 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.918 - 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117272997\)
\(L(\frac12)\) \(\approx\) \(1.117272997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 + 0.945i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
good2 \( 1 + (-0.723 - 0.690i)T^{2} \)
5 \( 1 + (-0.580 - 0.814i)T^{2} \)
11 \( 1 + (-0.580 - 0.814i)T^{2} \)
13 \( 1 + (-0.283 + 0.0270i)T + (0.981 - 0.189i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.514 - 0.404i)T + (0.235 + 0.971i)T^{2} \)
23 \( 1 + (0.786 - 0.618i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.975 + 1.37i)T + (-0.327 - 0.945i)T^{2} \)
37 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.0475 + 0.998i)T^{2} \)
43 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.142 + 0.989i)T^{2} \)
53 \( 1 + (0.888 + 0.458i)T^{2} \)
59 \( 1 + (-0.981 - 0.189i)T^{2} \)
61 \( 1 + (-0.581 + 0.299i)T + (0.580 - 0.814i)T^{2} \)
71 \( 1 + (0.888 + 0.458i)T^{2} \)
73 \( 1 + (-0.0800 - 0.0514i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.481 + 1.05i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.580 - 0.814i)T^{2} \)
89 \( 1 + (0.786 + 0.618i)T^{2} \)
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

−9.776933944463504580523027091575, −8.683024999899846632015865830418, −8.141651921892788341931932571306, −7.36006045791763600639361961449, −6.52977082030443650437628939901, −5.94291212033901878226523496944, −4.99759865842001590617222095706, −3.44161634050503089808069697569, −2.61441197391566683877555750009, −1.62770543399636193487093365585, 1.04175287625910126554892332434, 2.72427491532648352120425302680, 3.71491341366944839448041065418, 4.74139860415077052985670435066, 5.44840771887558812737015518502, 6.51182707500781642507877099407, 6.95410332262870034476776168258, 8.149021673779199580063803302222, 9.124350167961143612220664146260, 9.984456685407050168809665346475

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