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Properties

Label 2-414-9.7-c1-0-2
Degree $2$
Conductor $414$
Sign $0.671 - 0.740i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (0.5 − 0.866i)2-s + (−1.20 − 1.24i)3-s + (−0.499 − 0.866i)4-s + (1.39 + 2.40i)5-s + (−1.68 + 0.416i)6-s + (−1.77 + 3.07i)7-s − 0.999·8-s + (−0.114 + 2.99i)9-s + 2.78·10-s + (−2.05 + 3.55i)11-s + (−0.480 + 1.66i)12-s + (0.851 + 1.47i)13-s + (1.77 + 3.07i)14-s + (1.33 − 4.63i)15-s + (−0.5 + 0.866i)16-s − 1.77·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.693 − 0.720i)3-s + (−0.249 − 0.433i)4-s + (0.622 + 1.07i)5-s + (−0.686 + 0.169i)6-s + (−0.671 + 1.16i)7-s − 0.353·8-s + (−0.0382 + 0.999i)9-s + 0.879·10-s + (−0.618 + 1.07i)11-s + (−0.138 + 0.480i)12-s + (0.236 + 0.408i)13-s + (0.474 + 0.822i)14-s + (0.345 − 1.19i)15-s + (−0.125 + 0.216i)16-s − 0.429·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.671 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908517 + 0.402674i\)
\(L(\frac12)\) \(\approx\) \(0.908517 + 0.402674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1.20 + 1.24i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.39 - 2.40i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.77 - 3.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.05 - 3.55i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.851 - 1.47i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.77T + 17T^{2} \)
19 \( 1 + 3.17T + 19T^{2} \)
29 \( 1 + (-3.55 + 6.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.890 + 1.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.29T + 37T^{2} \)
41 \( 1 + (-4.67 - 8.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.80 - 6.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.11 + 8.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (-2.17 - 3.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.83 - 8.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.446 + 0.773i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.82T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + (5.37 - 9.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.68 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + (-6.33 + 10.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.52887443152215204577457569912, −10.46094402986710009230675763774, −9.938856722748756852153150316909, −8.754801332590239880235412440639, −7.31778361780889356827921328103, −6.34887512488573022013059003925, −5.85182748814721487187985973902, −4.57257029561012464244002086524, −2.69075626789017502521112699740, −2.11741945716828932068899389933, 0.60596705401694030222323140460, 3.37852302479900977744399803516, 4.42529551869736252311992880855, 5.35991583321493418744819710552, 6.10695468088639019735629424075, 7.11376203342088578537275063503, 8.552430271019050687990940366899, 9.147288569647159716787506010871, 10.40096920100720606749158068074, 10.76827135373722238967619670166

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