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Properties

Label 2-1407-1407.1046-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.495 - 0.868i$
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  + (0.0475 + 0.998i)3-s + (0.654 + 0.755i)4-s + (−0.235 + 0.971i)7-s + (−0.995 + 0.0950i)9-s + (−0.723 + 0.690i)12-s + (0.601 − 0.573i)13-s + (−0.142 + 0.989i)16-s + (−0.833 + 0.380i)19-s + (−0.981 − 0.189i)21-s + (0.723 − 0.690i)25-s + (−0.142 − 0.989i)27-s + (−0.888 + 0.458i)28-s + (−0.273 + 0.0801i)31-s + (−0.723 − 0.690i)36-s + 0.471·37-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)3-s + (0.654 + 0.755i)4-s + (−0.235 + 0.971i)7-s + (−0.995 + 0.0950i)9-s + (−0.723 + 0.690i)12-s + (0.601 − 0.573i)13-s + (−0.142 + 0.989i)16-s + (−0.833 + 0.380i)19-s + (−0.981 − 0.189i)21-s + (0.723 − 0.690i)25-s + (−0.142 − 0.989i)27-s + (−0.888 + 0.458i)28-s + (−0.273 + 0.0801i)31-s + (−0.723 − 0.690i)36-s + 0.471·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.495 - 0.868i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1046, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.495 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.183450526\)
\(L(\frac12)\) \(\approx\) \(1.183450526\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0475 - 0.998i)T \)
7 \( 1 + (0.235 - 0.971i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
good2 \( 1 + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (-0.723 + 0.690i)T^{2} \)
11 \( 1 + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.601 + 0.573i)T + (0.0475 - 0.998i)T^{2} \)
17 \( 1 + (0.928 + 0.371i)T^{2} \)
19 \( 1 + (0.833 - 0.380i)T + (0.654 - 0.755i)T^{2} \)
23 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 - 0.471T + T^{2} \)
41 \( 1 + (-0.928 - 0.371i)T^{2} \)
43 \( 1 + (-0.425 - 0.368i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.995 + 0.0950i)T^{2} \)
53 \( 1 + (0.928 - 0.371i)T^{2} \)
59 \( 1 + (-0.888 - 0.458i)T^{2} \)
61 \( 1 + (0.252 + 1.75i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.786 + 0.618i)T^{2} \)
73 \( 1 + (-0.459 - 0.584i)T + (-0.235 + 0.971i)T^{2} \)
79 \( 1 + (-1.34 - 1.40i)T + (-0.0475 + 0.998i)T^{2} \)
83 \( 1 + (0.235 + 0.971i)T^{2} \)
89 \( 1 + (-0.995 - 0.0950i)T^{2} \)
97 \( 1 + (-0.0475 + 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.03556121898895253798020872248, −9.124829013507530188699706211854, −8.440017060549855346215912424264, −7.912559314109102427222363076878, −6.53854166822352545006200909697, −5.98271745276184962668107669916, −4.96110609981475951132832365844, −3.88555750878878931281539155653, −3.08697451849691245239205520612, −2.24251869417704563757494318170, 0.994661365773624088112005050129, 2.01701578510686864303298177414, 3.15770322703875666035947272411, 4.41118797723966630045948549101, 5.61546771533990441923493872425, 6.40509324708691273605974834768, 6.97412024144656934730908296510, 7.58712067086576888922020397489, 8.669812354803385853622010043068, 9.437963170421464663863071037769

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