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Properties

Label 2-414-23.13-c1-0-7
Degree $2$
Conductor $414$
Sign $0.355 + 0.934i$
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  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (1.37 − 3.01i)5-s + (3.66 + 1.07i)7-s + (0.841 − 0.540i)8-s + (−3.18 + 0.934i)10-s + (−0.230 + 0.265i)11-s + (−0.367 + 0.107i)13-s + (−1.58 − 3.47i)14-s + (−0.959 − 0.281i)16-s + (1.03 + 7.20i)17-s + (0.592 − 4.11i)19-s + (2.79 + 1.79i)20-s + 0.351·22-s + (0.368 − 4.78i)23-s + ⋯
L(s)  = 1  + (−0.463 − 0.534i)2-s + (−0.0711 + 0.494i)4-s + (0.616 − 1.34i)5-s + (1.38 + 0.407i)7-s + (0.297 − 0.191i)8-s + (−1.00 + 0.295i)10-s + (−0.0694 + 0.0801i)11-s + (−0.101 + 0.0299i)13-s + (−0.424 − 0.929i)14-s + (−0.239 − 0.0704i)16-s + (0.251 + 1.74i)17-s + (0.135 − 0.944i)19-s + (0.623 + 0.400i)20-s + 0.0750·22-s + (0.0768 − 0.997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.355 + 0.934i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12179 - 0.773670i\)
\(L(\frac12)\) \(\approx\) \(1.12179 - 0.773670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 \)
23 \( 1 + (-0.368 + 4.78i)T \)
good5 \( 1 + (-1.37 + 3.01i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-3.66 - 1.07i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (0.230 - 0.265i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.367 - 0.107i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-1.03 - 7.20i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.592 + 4.11i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.637 + 4.43i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-3.65 + 2.35i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (3.04 + 6.65i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (2.92 - 6.40i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-7.42 - 4.76i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + (-2.43 - 0.714i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-7.57 + 2.22i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (10.6 - 6.84i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (3.67 + 4.23i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (0.217 + 0.250i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.0995 - 0.692i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (15.4 - 4.54i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-1.92 - 4.21i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-7.41 - 4.76i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (4.66 - 10.2i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.05664505773346544865331655385, −10.11652805914585304670259133722, −9.106885862686098912247934743564, −8.458321416502579087688446611301, −7.84060095480095625333512848137, −6.13852351360766783766045056686, −5.03456914856325717346165119352, −4.28453947283741102572764993122, −2.27309091049374158579589047337, −1.25081170583059260511368707525, 1.66168538140030909842434082430, 3.14422051027192550446337481931, 4.85897399748692285447213722013, 5.75320437521594880543416086734, 7.02904602434820456265880768151, 7.45762253803037307689418973754, 8.529410576380678624271154029943, 9.730959523515980878891940232699, 10.38928777729531834242925711837, 11.19457310623511958031905153747

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