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Properties

Label 2-1407-1407.1343-c0-0-1
Degree $2$
Conductor $1407$
Sign $-0.687 - 0.725i$
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  + (0.142 − 0.989i)3-s + (−0.841 − 0.540i)4-s + (−0.959 + 0.281i)7-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)12-s + (−1.25 + 1.45i)13-s + (0.415 + 0.909i)16-s + (−0.557 − 1.89i)19-s + (0.142 + 0.989i)21-s + (−0.654 + 0.755i)25-s + (−0.415 + 0.909i)27-s + (0.959 + 0.281i)28-s + (−0.544 − 0.627i)31-s + (0.654 + 0.755i)36-s − 1.30·37-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (−0.841 − 0.540i)4-s + (−0.959 + 0.281i)7-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)12-s + (−1.25 + 1.45i)13-s + (0.415 + 0.909i)16-s + (−0.557 − 1.89i)19-s + (0.142 + 0.989i)21-s + (−0.654 + 0.755i)25-s + (−0.415 + 0.909i)27-s + (0.959 + 0.281i)28-s + (−0.544 − 0.627i)31-s + (0.654 + 0.755i)36-s − 1.30·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.687 - 0.725i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.687 - 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08464427680\)
\(L(\frac12)\) \(\approx\) \(0.08464427680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
good2 \( 1 + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.557 + 1.89i)T + (-0.841 + 0.540i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + 1.30T + T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.817 - 1.27i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (1.65 + 0.755i)T + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + 0.284T + T^{2} \)
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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

−9.252869715880150892089841273856, −8.698097846137928602264105404820, −7.43837680710987481496630116006, −6.82938169597923597446321280408, −6.06900579133574262930589414267, −5.11729572946754567066471692545, −4.17684116086377007474991147468, −2.85898697849571174266999869745, −1.84109428332396703730308276062, −0.06537213020839035265452100655, 2.64925355023899908962525221287, 3.56505252558323371265404115292, 4.11342916845776010142116149222, 5.25278276417589682966494111337, 5.82182098173650191288871347621, 7.22595554304796265059424380143, 8.051271755258336607608711314529, 8.684616282433413280232838321488, 9.610466378403734226726213993726, 10.21630002502648095913557427417

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