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Properties

Label 2-1407-1407.65-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.198 - 0.980i$
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  + (0.235 + 0.971i)3-s + (0.415 + 0.909i)4-s + (0.928 − 0.371i)7-s + (−0.888 + 0.458i)9-s + (−0.786 + 0.618i)12-s + (−1.32 + 1.04i)13-s + (−0.654 + 0.755i)16-s + (1.21 − 0.782i)19-s + (0.580 + 0.814i)21-s + (−0.786 + 0.618i)25-s + (−0.654 − 0.755i)27-s + (0.723 + 0.690i)28-s + (0.186 − 1.29i)31-s + (−0.786 − 0.618i)36-s + 1.85·37-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)3-s + (0.415 + 0.909i)4-s + (0.928 − 0.371i)7-s + (−0.888 + 0.458i)9-s + (−0.786 + 0.618i)12-s + (−1.32 + 1.04i)13-s + (−0.654 + 0.755i)16-s + (1.21 − 0.782i)19-s + (0.580 + 0.814i)21-s + (−0.786 + 0.618i)25-s + (−0.654 − 0.755i)27-s + (0.723 + 0.690i)28-s + (0.186 − 1.29i)31-s + (−0.786 − 0.618i)36-s + 1.85·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.198 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301562219\)
\(L(\frac12)\) \(\approx\) \(1.301562219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 - 0.971i)T \)
7 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
good2 \( 1 + (-0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.786 - 0.618i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (1.32 - 1.04i)T + (0.235 - 0.971i)T^{2} \)
17 \( 1 + (0.327 - 0.945i)T^{2} \)
19 \( 1 + (-1.21 + 0.782i)T + (0.415 - 0.909i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 - 1.85T + T^{2} \)
41 \( 1 + (0.327 - 0.945i)T^{2} \)
43 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.888 - 0.458i)T^{2} \)
53 \( 1 + (0.327 + 0.945i)T^{2} \)
59 \( 1 + (-0.723 + 0.690i)T^{2} \)
61 \( 1 + (0.947 + 1.09i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.981 - 0.189i)T^{2} \)
73 \( 1 + (0.642 - 0.123i)T + (0.928 - 0.371i)T^{2} \)
79 \( 1 + (1.45 - 1.14i)T + (0.235 - 0.971i)T^{2} \)
83 \( 1 + (-0.928 - 0.371i)T^{2} \)
89 \( 1 + (0.888 + 0.458i)T^{2} \)
97 \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.673745954053677629957996167552, −9.411248000746558315938899678587, −8.269548126755332404895524349106, −7.65681978104949067327888974775, −7.01934924912448395158227121164, −5.67827879674361273248424321400, −4.61769449897863977956400671147, −4.18019445122809259881515802533, −2.99750098039201410855064585923, −2.10899828354984616984035745178, 1.11392400187240539496194421983, 2.17148596396321701370879588218, 3.00492158594603584186232033883, 4.74682366629406041691169455758, 5.55029902800238969233472234012, 6.11811059138842954938270259601, 7.34151592989483353172910412991, 7.68175488882404290705143121615, 8.579878994437678103042547426712, 9.616190387819303710990464038580

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