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Properties

Label 2-42e2-28.27-c1-0-94
Degree $2$
Conductor $1764$
Sign $-0.944 + 0.327i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s − 2.82i·5-s − 2.82i·8-s + (−2.00 − 3.46i)10-s − 5.65i·11-s + 5.19i·13-s + (−2.00 − 3.46i)16-s − 2.82i·17-s − 5·19-s + (−4.89 − 2.82i)20-s + (−4.00 − 6.92i)22-s + 2.82i·23-s − 3.00·25-s + (3.67 + 6.36i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s − 1.26i·5-s − 0.999i·8-s + (−0.632 − 1.09i)10-s − 1.70i·11-s + 1.44i·13-s + (−0.500 − 0.866i)16-s − 0.685i·17-s − 1.14·19-s + (−1.09 − 0.632i)20-s + (−0.852 − 1.47i)22-s + 0.589i·23-s − 0.600·25-s + (0.720 + 1.24i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.944 + 0.327i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.498710605\)
\(L(\frac12)\) \(\approx\) \(2.498710605\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.82iT - 5T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 5.19iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 - 4.89T + 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 1.73iT - 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.972936478397367090983615351884, −8.432222046197361990088234395253, −7.18569321772295993410276503781, −6.20852886570848871896425790395, −5.61850897855876204226254071666, −4.64066978945367483892715451175, −4.09986988919614786639420109537, −2.99346417852423039442607799818, −1.77458789953993893363670907177, −0.67024818240237012750038586797, 2.15517206375599919105648209805, 2.84500704331617011819116857711, 3.90848446089770691695777244188, 4.67111419823387410073788205638, 5.73914660609603875801311357219, 6.47169478361133191405892828482, 7.14032325061611360501246968241, 7.77209044044676253754550677111, 8.562798733159473732910819959442, 9.873592605453380239754435164569

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