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Properties

Label 2-414-23.8-c1-0-7
Degree $2$
Conductor $414$
Sign $0.429 + 0.903i$
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  + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.357 + 0.412i)5-s + (−3.96 − 2.54i)7-s + (0.415 − 0.909i)8-s + (−0.459 + 0.295i)10-s + (−0.698 − 4.86i)11-s + (2.05 − 1.31i)13-s + (3.08 − 3.55i)14-s + (0.841 + 0.540i)16-s + (1.58 − 0.466i)17-s + (−1.11 − 0.328i)19-s + (−0.226 − 0.496i)20-s + 4.91·22-s + (−4.79 − 0.102i)23-s + ⋯
L(s)  = 1  + (−0.100 + 0.699i)2-s + (−0.479 − 0.140i)4-s + (0.159 + 0.184i)5-s + (−1.49 − 0.961i)7-s + (0.146 − 0.321i)8-s + (−0.145 + 0.0933i)10-s + (−0.210 − 1.46i)11-s + (0.568 − 0.365i)13-s + (0.823 − 0.950i)14-s + (0.210 + 0.135i)16-s + (0.385 − 0.113i)17-s + (−0.256 − 0.0753i)19-s + (−0.0507 − 0.111i)20-s + 1.04·22-s + (−0.999 − 0.0213i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.429 + 0.903i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.429 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658364 - 0.415965i\)
\(L(\frac12)\) \(\approx\) \(0.658364 - 0.415965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 \)
23 \( 1 + (4.79 + 0.102i)T \)
good5 \( 1 + (-0.357 - 0.412i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (3.96 + 2.54i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.698 + 4.86i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-2.05 + 1.31i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-1.58 + 0.466i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (1.11 + 0.328i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-2.00 + 0.588i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (2.66 - 5.84i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-1.83 + 2.11i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.96 + 2.26i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (4.45 + 9.76i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + (2.93 + 1.88i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (2.76 - 1.77i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (2.18 - 4.77i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-0.0899 + 0.625i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.876 - 6.09i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (10.6 + 3.13i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-3.05 + 1.96i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-3.19 + 3.68i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-0.310 - 0.680i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-3.07 - 3.54i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.54532506116931731774760321678, −10.34964926586135918542176737157, −9.117049702598158945789574726594, −8.283643198680707255384339807682, −7.18918296880368271864560296174, −6.31911000237816795500481727396, −5.67467646113373514838151847222, −3.98821892882497676093313309274, −3.12653140026203534558682616606, −0.51191606278133008284485566729, 1.93326906455707473892118399088, 3.11394393996434643612076072302, 4.32392360245282650281510209695, 5.65231889868258755238514955444, 6.56548697261322149489997307095, 7.82555364611835234922478654518, 9.061917304883381075132275319321, 9.607372132569289328525389792738, 10.25761951449186222969615132470, 11.51793361714342850469908198321

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