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Properties

Label 2-406-7.2-c1-0-1
Degree $2$
Conductor $406$
Sign $-0.981 + 0.189i$
Analytic cond. $3.24192$
Root an. cond. $1.80053$
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 + (1.22 + 2.11i)3-s + (−0.499 − 0.866i)4-s + (−2.07 + 3.58i)5-s − 2.44·6-s + (−0.335 − 2.62i)7-s + 0.999·8-s + (−1.48 + 2.56i)9-s + (−2.07 − 3.58i)10-s + (2.35 + 4.07i)11-s + (1.22 − 2.11i)12-s − 1.91·13-s + (2.44 + 1.02i)14-s − 10.1·15-s + (−0.5 + 0.866i)16-s + (−1.38 − 2.39i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.705 + 1.22i)3-s + (−0.249 − 0.433i)4-s + (−0.926 + 1.60i)5-s − 0.997·6-s + (−0.126 − 0.991i)7-s + 0.353·8-s + (−0.494 + 0.856i)9-s + (−0.655 − 1.13i)10-s + (0.709 + 1.22i)11-s + (0.352 − 0.610i)12-s − 0.529·13-s + (0.652 + 0.272i)14-s − 2.61·15-s + (−0.125 + 0.216i)16-s + (−0.335 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(406\)    =    \(2 \cdot 7 \cdot 29\)
Sign: $-0.981 + 0.189i$
Analytic conductor: \(3.24192\)
Root analytic conductor: \(1.80053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{406} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 406,\ (\ :1/2),\ -0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0968401 - 1.01266i\)
\(L(\frac12)\) \(\approx\) \(0.0968401 - 1.01266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.335 + 2.62i)T \)
29 \( 1 + T \)
good3 \( 1 + (-1.22 - 2.11i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.07 - 3.58i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.91T + 13T^{2} \)
17 \( 1 + (1.38 + 2.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.02 - 3.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.08 + 1.88i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-0.484 - 0.839i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.736 + 1.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 7.84T + 43T^{2} \)
47 \( 1 + (5.86 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.39 - 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.70 + 2.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.70 - 9.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.13 - 1.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 + (-6.76 - 11.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.23 + 3.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.06T + 83T^{2} \)
89 \( 1 + (6.63 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−11.26770205015887708711300749034, −10.52962263557898750222433875693, −9.937529532772073124572078736573, −9.182942402849601028337868712872, −7.80597672794756849114132495696, −7.26883366607739401734244988028, −6.42970338305873372426958496215, −4.42041620009662559913989144167, −4.03486388851155453730492737484, −2.76168929499677774118702486547, 0.70051148543055128588751447058, 2.00982342812142428317385192032, 3.38105933494367152194068337580, 4.68550538978842640700000562046, 6.05573670298281316794624774407, 7.44712806956378087832380394079, 8.314799367345565694531973752283, 8.787405507091647794202302976346, 9.318790455057132561376656923069, 11.19641514310985602532760919392

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