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Properties

Label 2-693-231.62-c0-0-3
Degree $2$
Conductor $693$
Sign $0.997 - 0.0746i$
Analytic cond. $0.345852$
Root an. cond. $0.588091$
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  + (0.734 + 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (0.329 − 1.01i)8-s + (−0.987 − 0.156i)11-s + (0.863 + 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s + 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (0.0966 + 0.297i)29-s − 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (0.0274 + 0.173i)44-s + ⋯
L(s)  = 1  + (0.734 + 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (0.329 − 1.01i)8-s + (−0.987 − 0.156i)11-s + (0.863 + 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s + 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (0.0966 + 0.297i)29-s − 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (0.0274 + 0.173i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.997 - 0.0746i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.997 - 0.0746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369966531\)
\(L(\frac12)\) \(\approx\) \(1.369966531\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.987 + 0.156i)T \)
good2 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.78iT - T^{2} \)
29 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−10.77254866478324440150655696932, −9.880802234945932004551327014305, −8.951383171827479867588831345357, −7.66011984492660116394288667895, −7.33206266450069660727028453804, −5.91119475130221063048095863582, −5.31822418556367670268786928249, −4.47994157932479598312737273952, −3.36715407851992918063285398973, −1.59401119483080453256658171939, 2.07150309078283174174464477932, 2.93546457510629599122432856361, 4.35470421651887605486003172613, 4.89208157100979186150687756097, 5.90320555195122580458230614250, 7.25759176749485244428223225227, 8.297957438868835182634586480974, 8.586522719768385252551770152589, 10.19853779842858046770062617680, 10.72356197103482372967016379335

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