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Properties

Label 2-414-9.7-c1-0-1
Degree $2$
Conductor $414$
Sign $-0.989 - 0.144i$
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 + (−1.66 + 0.473i)3-s + (−0.499 − 0.866i)4-s + (0.274 + 0.476i)5-s + (0.423 − 1.67i)6-s + (0.708 − 1.22i)7-s + 0.999·8-s + (2.55 − 1.57i)9-s − 0.549·10-s + (−2.68 + 4.64i)11-s + (1.24 + 1.20i)12-s + (2.46 + 4.27i)13-s + (0.708 + 1.22i)14-s + (−0.683 − 0.663i)15-s + (−0.5 + 0.866i)16-s − 5.10·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.961 + 0.273i)3-s + (−0.249 − 0.433i)4-s + (0.122 + 0.212i)5-s + (0.172 − 0.685i)6-s + (0.267 − 0.463i)7-s + 0.353·8-s + (0.850 − 0.525i)9-s − 0.173·10-s + (−0.809 + 1.40i)11-s + (0.358 + 0.348i)12-s + (0.684 + 1.18i)13-s + (0.189 + 0.327i)14-s + (−0.176 − 0.171i)15-s + (−0.125 + 0.216i)16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.989 - 0.144i$
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: \((2,\ 414,\ (\ :1/2),\ -0.989 - 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0334071 + 0.460430i\)
\(L(\frac12)\) \(\approx\) \(0.0334071 + 0.460430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 0.473i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.274 - 0.476i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.708 + 1.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.68 - 4.64i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.46 - 4.27i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.10T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
29 \( 1 + (1.63 - 2.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.64 + 4.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.38T + 37T^{2} \)
41 \( 1 + (-5.82 - 10.0i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.67 - 9.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.98 - 3.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.36T + 53T^{2} \)
59 \( 1 + (-5.65 - 9.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.93 + 8.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.36 + 7.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + (-1.72 + 2.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.00 + 3.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.39T + 89T^{2} \)
97 \( 1 + (7.05 - 12.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.27906272706254433989254958176, −10.76317691564844776921531920118, −9.900515337799107149141312744430, −9.023888952109197690162242419965, −7.79239829000234865967066248096, −6.74147660751211614427823004884, −6.30606422758513339200580047479, −4.75506322210786798720213129544, −4.33874576746438107105843356221, −1.86916581949056982266069648125, 0.36717894944553653146920656347, 2.06673177915636443911724598170, 3.61518540696669415574876752478, 5.15902627136573332065047741658, 5.79053159167471257705595669488, 7.02664410751793459582939586684, 8.358395129608597188974132700566, 8.764413669968963394727187030012, 10.33773707021606117650980601183, 10.85569582659191786137139661340

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