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Properties

Label 2-403-403.36-c1-0-0
Degree $2$
Conductor $403$
Sign $-0.904 + 0.426i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + (−0.700 + 0.404i)2-s + 1.07·3-s + (−0.672 + 1.16i)4-s + (−3.37 + 1.95i)5-s + (−0.750 + 0.433i)6-s + (2.06 − 1.19i)7-s − 2.70i·8-s − 1.85·9-s + (1.57 − 2.73i)10-s + (0.548 + 0.316i)11-s + (−0.720 + 1.24i)12-s + (−2.35 + 2.73i)13-s + (−0.965 + 1.67i)14-s + (−3.61 + 2.08i)15-s + (−0.249 − 0.432i)16-s + (−1.34 − 2.32i)17-s + ⋯
L(s)  = 1  + (−0.495 + 0.286i)2-s + 0.618·3-s + (−0.336 + 0.582i)4-s + (−1.51 + 0.872i)5-s + (−0.306 + 0.176i)6-s + (0.780 − 0.450i)7-s − 0.957i·8-s − 0.617·9-s + (0.499 − 0.864i)10-s + (0.165 + 0.0954i)11-s + (−0.207 + 0.360i)12-s + (−0.652 + 0.757i)13-s + (−0.257 + 0.446i)14-s + (−0.934 + 0.539i)15-s + (−0.0624 − 0.108i)16-s + (−0.325 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0521800 - 0.233097i\)
\(L(\frac12)\) \(\approx\) \(0.0521800 - 0.233097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.35 - 2.73i)T \)
31 \( 1 + (0.0308 - 5.56i)T \)
good2 \( 1 + (0.700 - 0.404i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 1.07T + 3T^{2} \)
5 \( 1 + (3.37 - 1.95i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.06 + 1.19i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.548 - 0.316i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.34 + 2.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.14 - 1.81i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.45 + 4.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.82 + 6.61i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 - 6.64iT - 37T^{2} \)
41 \( 1 + (5.47 + 3.16i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.47 - 2.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.05iT - 47T^{2} \)
53 \( 1 + (-3.64 - 6.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.44 - 4.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.21 + 3.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 5.98i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.91iT - 71T^{2} \)
73 \( 1 + (-5.01 + 2.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.16 - 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.99 - 5.77i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.10 - 1.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.2 + 8.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.70593829710752703726350887255, −10.98016395548180442197939130511, −9.817238653044028800672474087606, −8.633541170253163081413393802616, −8.125485718093541981263821651726, −7.41899901332004023058007258171, −6.66132739267232301480315524535, −4.50558562182784043679445034922, −3.86294825980011440954613327940, −2.66766188522061561029036521518, 0.16568042954027369845564054520, 1.97979310693301790591249121354, 3.64579691975035311942438686978, 4.81440195661565042970113548158, 5.61984703954379007217936939058, 7.53377237275796195424948801834, 8.349820619153258469219013050101, 8.664155593479401420869301944660, 9.581989414197676326044569174848, 10.96460117054836210370937492036

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