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Properties

Label 2-693-231.167-c0-0-0
Degree $2$
Conductor $693$
Sign $0.190 - 0.981i$
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  + (−0.0966 + 0.297i)2-s + (0.729 + 0.530i)4-s + (−0.587 + 0.809i)7-s + (−0.481 + 0.349i)8-s + (−0.453 + 0.891i)11-s + (−0.183 − 0.253i)14-s + (0.221 + 0.680i)16-s + (−0.221 − 0.221i)22-s − 1.97i·23-s + (0.809 − 0.587i)25-s + (−0.857 + 0.278i)28-s + (1.44 + 1.04i)29-s − 0.819·32-s + (−0.951 − 0.690i)37-s − 0.618i·43-s + (−0.803 + 0.409i)44-s + ⋯
L(s)  = 1  + (−0.0966 + 0.297i)2-s + (0.729 + 0.530i)4-s + (−0.587 + 0.809i)7-s + (−0.481 + 0.349i)8-s + (−0.453 + 0.891i)11-s + (−0.183 − 0.253i)14-s + (0.221 + 0.680i)16-s + (−0.221 − 0.221i)22-s − 1.97i·23-s + (0.809 − 0.587i)25-s + (−0.857 + 0.278i)28-s + (1.44 + 1.04i)29-s − 0.819·32-s + (−0.951 − 0.690i)37-s − 0.618i·43-s + (−0.803 + 0.409i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.190 - 0.981i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.190 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9595192121\)
\(L(\frac12)\) \(\approx\) \(0.9595192121\)
\(L(1)\) 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 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.453 - 0.891i)T \)
good2 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.97iT - T^{2} \)
29 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.69 - 0.550i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.863 - 0.280i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−10.68151005833140564249556600798, −10.16942601512139081162810224803, −8.792458705505487334432440055939, −8.453802646447462222938220823839, −7.13398731649512979840997806798, −6.67387597092873804679454244932, −5.64094936024933737088868088790, −4.49216267179879403719268958528, −3.01177590942118824597178671337, −2.27602869291818794927641599057, 1.16817165119937264475905190244, 2.79757146720682771443586220879, 3.65045408247538775816267399231, 5.16992503337588871658804787849, 6.09563358329533034294729899154, 6.92490468638593305273735200778, 7.74804915904844083266723294502, 8.910670978511422176849034732771, 9.951886311050152722803010337681, 10.37594545041502237276629913250

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