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Properties

Label 2-414-23.12-c1-0-2
Degree $2$
Conductor $414$
Sign $0.897 - 0.440i$
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  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (3.68 − 1.08i)5-s + (3.32 + 3.84i)7-s + (0.142 + 0.989i)8-s + (−2.51 + 2.89i)10-s + (−3.79 − 2.44i)11-s + (2.18 − 2.51i)13-s + (−4.87 − 1.43i)14-s + (−0.654 − 0.755i)16-s + (0.0640 + 0.140i)17-s + (−2.18 + 4.77i)19-s + (0.545 − 3.79i)20-s + 4.51·22-s + (−2.86 − 3.84i)23-s + ⋯
L(s)  = 1  + (−0.594 + 0.382i)2-s + (0.207 − 0.454i)4-s + (1.64 − 0.483i)5-s + (1.25 + 1.45i)7-s + (0.0503 + 0.349i)8-s + (−0.794 + 0.916i)10-s + (−1.14 − 0.736i)11-s + (0.604 − 0.697i)13-s + (−1.30 − 0.382i)14-s + (−0.163 − 0.188i)16-s + (0.0155 + 0.0339i)17-s + (−0.500 + 1.09i)19-s + (0.122 − 0.848i)20-s + 0.963·22-s + (−0.597 − 0.802i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.897 - 0.440i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.897 - 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42575 + 0.331228i\)
\(L(\frac12)\) \(\approx\) \(1.42575 + 0.331228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 \)
23 \( 1 + (2.86 + 3.84i)T \)
good5 \( 1 + (-3.68 + 1.08i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-3.32 - 3.84i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (3.79 + 2.44i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-2.18 + 2.51i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-0.0640 - 0.140i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.18 - 4.77i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.17 - 4.76i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.180 + 1.25i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (4.27 + 1.25i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-2.12 + 0.623i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.375 - 2.61i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 1.10T + 47T^{2} \)
53 \( 1 + (-2.80 - 3.23i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-2.11 + 2.43i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.87 + 13.0i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-8.81 + 5.66i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (11.9 - 7.70i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-2.57 + 5.64i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (1.91 - 2.20i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-2.23 - 0.657i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.783 + 5.45i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (12.6 - 3.70i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.91328805840598607590747381531, −10.37046230092372454380434483068, −9.295646683496182361386618950141, −8.398638349085013842990452819601, −8.145822288990946573579236243351, −6.22152810778544130555094179780, −5.62450768612957216780201896063, −5.06475555695715562241372772152, −2.57146162595670140306008741082, −1.61112053321897856493381882305, 1.50052780177423257095707725802, 2.43934557002273905350738002641, 4.22944486380874127131720603648, 5.33098711455556988427320399872, 6.70429099236629423471123441921, 7.43115732046095178610735921413, 8.483103377134792320284276262926, 9.621827156324139099505166417301, 10.35982375684844620657511494606, 10.78123329651835154670016932707

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