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Properties

Label 2-351-39.8-c1-0-15
Degree $2$
Conductor $351$
Sign $0.104 + 0.994i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (1.35 − 1.35i)2-s − 1.66i·4-s + (2.02 − 2.02i)5-s + (0.0947 − 0.0947i)7-s + (0.448 + 0.448i)8-s − 5.48i·10-s + (1.48 + 1.48i)11-s + (−3.32 + 1.40i)13-s − 0.256i·14-s + 4.55·16-s − 4.69·17-s + (−4.59 − 4.59i)19-s + (−3.37 − 3.37i)20-s + 4.01·22-s + 3.79·23-s + ⋯
L(s)  = 1  + (0.957 − 0.957i)2-s − 0.834i·4-s + (0.905 − 0.905i)5-s + (0.0358 − 0.0358i)7-s + (0.158 + 0.158i)8-s − 1.73i·10-s + (0.447 + 0.447i)11-s + (−0.921 + 0.388i)13-s − 0.0686i·14-s + 1.13·16-s − 1.13·17-s + (−1.05 − 1.05i)19-s + (−0.755 − 0.755i)20-s + 0.856·22-s + 0.791·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.104 + 0.994i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.104 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80270 - 1.62272i\)
\(L(\frac12)\) \(\approx\) \(1.80270 - 1.62272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (3.32 - 1.40i)T \)
good2 \( 1 + (-1.35 + 1.35i)T - 2iT^{2} \)
5 \( 1 + (-2.02 + 2.02i)T - 5iT^{2} \)
7 \( 1 + (-0.0947 + 0.0947i)T - 7iT^{2} \)
11 \( 1 + (-1.48 - 1.48i)T + 11iT^{2} \)
17 \( 1 + 4.69T + 17T^{2} \)
19 \( 1 + (4.59 + 4.59i)T + 19iT^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 - 6.75iT - 29T^{2} \)
31 \( 1 + (2.65 + 2.65i)T + 31iT^{2} \)
37 \( 1 + (-4.05 + 4.05i)T - 37iT^{2} \)
41 \( 1 + (1.48 - 1.48i)T - 41iT^{2} \)
43 \( 1 - 7.45iT - 43T^{2} \)
47 \( 1 + (-5.23 - 5.23i)T + 47iT^{2} \)
53 \( 1 - 4.43iT - 53T^{2} \)
59 \( 1 + (0.799 + 0.799i)T + 59iT^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + (4.06 + 4.06i)T + 67iT^{2} \)
71 \( 1 + (-5.81 + 5.81i)T - 71iT^{2} \)
73 \( 1 + (8.62 - 8.62i)T - 73iT^{2} \)
79 \( 1 + 4.32T + 79T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 + (3.25 + 3.25i)T + 89iT^{2} \)
97 \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \)
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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

−11.34018314196044765970298140020, −10.67446052253138359553111330802, −9.404193786442339067325256687752, −8.923853677741291709109434231242, −7.33477239764128287034278779197, −6.09430120953824064299825649248, −4.79452933150881973527254892771, −4.46958036401204532080280680879, −2.69072886359732922034286176808, −1.66420659093730881451818463571, 2.29440529240334901810977157170, 3.76366083017954465790660757080, 4.98165274421633492030495782776, 6.04855721929365003887995566849, 6.57716843021987706485573262578, 7.50613680279334613096356603327, 8.770582253083414175596839237207, 10.04833918428445455312584113208, 10.62036936247190618464438649935, 11.88938358033360263296627121326

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