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Properties

Label 2-414-9.4-c1-0-13
Degree $2$
Conductor $414$
Sign $0.999 + 0.0342i$
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 + (−1.61 − 0.620i)3-s + (−0.499 + 0.866i)4-s + (0.536 − 0.928i)5-s + (−0.271 − 1.71i)6-s + (−0.121 − 0.211i)7-s − 0.999·8-s + (2.23 + 2.00i)9-s + 1.07·10-s + (−0.317 − 0.549i)11-s + (1.34 − 1.09i)12-s + (2.75 − 4.77i)13-s + (0.121 − 0.211i)14-s + (−1.44 + 1.16i)15-s + (−0.5 − 0.866i)16-s + 7.28·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.933 − 0.358i)3-s + (−0.249 + 0.433i)4-s + (0.239 − 0.415i)5-s + (−0.110 − 0.698i)6-s + (−0.0460 − 0.0798i)7-s − 0.353·8-s + (0.743 + 0.668i)9-s + 0.339·10-s + (−0.0956 − 0.165i)11-s + (0.388 − 0.314i)12-s + (0.764 − 1.32i)13-s + (0.0325 − 0.0564i)14-s + (−0.372 + 0.301i)15-s + (−0.125 − 0.216i)16-s + 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.999 + 0.0342i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.999 + 0.0342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29504 - 0.0222111i\)
\(L(\frac12)\) \(\approx\) \(1.29504 - 0.0222111i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.620i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.536 + 0.928i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.121 + 0.211i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.317 + 0.549i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.75 + 4.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.28T + 17T^{2} \)
19 \( 1 - 1.65T + 19T^{2} \)
29 \( 1 + (2.59 + 4.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.82 + 6.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.11T + 37T^{2} \)
41 \( 1 + (3.51 - 6.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.73 - 4.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.39 - 4.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
59 \( 1 + (5.47 - 9.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.02 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.121 - 0.211i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.03T + 71T^{2} \)
73 \( 1 + 0.0957T + 73T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.66 + 8.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.00T + 89T^{2} \)
97 \( 1 + (-4.59 - 7.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.38708697061839650489413569656, −10.34539761188497752808482788416, −9.475782798574690905257109885599, −7.966765995351588943832337041179, −7.62465847487759640547073426407, −6.07657212077122609564060286435, −5.73669137491936781931145743015, −4.70707223920306638131979762635, −3.26105249934799425870477991597, −1.03606227550314843133327818850, 1.41347394058441208997029240118, 3.21740010107713385842097365574, 4.31614438339273500131975055632, 5.39933899980433274353928770553, 6.26379468007091433896322705916, 7.21185259615735538057873337303, 8.806112774025514548471110765503, 9.777133568196975173795713841123, 10.46709754694212677444243467055, 11.20428029451783508925215700579

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