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Properties

Label 2-414-9.7-c1-0-14
Degree $2$
Conductor $414$
Sign $0.810 + 0.585i$
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 + (1.73 − 0.0627i)3-s + (−0.499 − 0.866i)4-s + (1.43 + 2.48i)5-s + (0.811 − 1.53i)6-s + (1.80 − 3.13i)7-s − 0.999·8-s + (2.99 − 0.217i)9-s + 2.86·10-s + (−2.96 + 5.14i)11-s + (−0.919 − 1.46i)12-s + (−1.07 − 1.86i)13-s + (−1.80 − 3.13i)14-s + (2.63 + 4.20i)15-s + (−0.5 + 0.866i)16-s + 1.69·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.999 − 0.0362i)3-s + (−0.249 − 0.433i)4-s + (0.640 + 1.11i)5-s + (0.331 − 0.624i)6-s + (0.683 − 1.18i)7-s − 0.353·8-s + (0.997 − 0.0724i)9-s + 0.906·10-s + (−0.895 + 1.55i)11-s + (−0.265 − 0.423i)12-s + (−0.298 − 0.517i)13-s + (−0.483 − 0.836i)14-s + (0.680 + 1.08i)15-s + (−0.125 + 0.216i)16-s + 0.409·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.810 + 0.585i$
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: \((2,\ 414,\ (\ :1/2),\ 0.810 + 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30971 - 0.747003i\)
\(L(\frac12)\) \(\approx\) \(2.30971 - 0.747003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0627i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.43 - 2.48i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.80 + 3.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.96 - 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.07 + 1.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.69T + 17T^{2} \)
19 \( 1 + 5.76T + 19T^{2} \)
29 \( 1 + (-3.61 + 6.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.78 - 3.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.75T + 37T^{2} \)
41 \( 1 + (6.02 + 10.4i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.667 - 1.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.92 - 5.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.70T + 53T^{2} \)
59 \( 1 + (-6.50 - 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.53 + 4.39i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.80 - 3.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 + (-0.721 + 1.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.15 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (1.61 - 2.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60157701931712745840763132659, −10.39964401661261410651010650199, −9.780242902138750866195828349819, −8.290163456042665575510018075869, −7.40933921261800619703002524660, −6.65027663969699466528130965828, −4.94628791586623620622673251994, −4.03036429513644087672201381908, −2.72178752541589522511375795264, −1.88895554314995649490598330062, 1.88551091773091280102064044679, 3.17329589530296970099717931838, 4.75939598639370095429492655710, 5.37493894489227644979711863202, 6.46491452369081958653841321029, 8.055266302773822936655866799889, 8.527556733442122631900641959969, 8.992080968923493757749158752734, 10.14117797702204816115233645922, 11.49308891545642635548080886192

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