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Properties

Label 2-403-403.6-c1-0-9
Degree $2$
Conductor $403$
Sign $0.279 + 0.960i$
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  + (−0.626 − 2.33i)2-s + 1.39i·3-s + (−3.33 + 1.92i)4-s + (−0.0315 − 0.117i)5-s + (3.25 − 0.872i)6-s + (−1.74 + 0.468i)7-s + (3.17 + 3.17i)8-s + 1.06·9-s + (−0.255 + 0.147i)10-s + (1.33 + 0.358i)11-s + (−2.68 − 4.65i)12-s + (3.37 − 1.25i)13-s + (2.18 + 3.79i)14-s + (0.163 − 0.0439i)15-s + (1.57 − 2.73i)16-s + (3.20 − 5.54i)17-s + ⋯
L(s)  = 1  + (−0.442 − 1.65i)2-s + 0.804i·3-s + (−1.66 + 0.963i)4-s + (−0.0140 − 0.0526i)5-s + (1.32 − 0.356i)6-s + (−0.660 + 0.177i)7-s + (1.12 + 1.12i)8-s + 0.353·9-s + (−0.0807 + 0.0465i)10-s + (0.403 + 0.108i)11-s + (−0.775 − 1.34i)12-s + (0.937 − 0.349i)13-s + (0.585 + 1.01i)14-s + (0.0423 − 0.0113i)15-s + (0.394 − 0.682i)16-s + (0.776 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831940 - 0.623965i\)
\(L(\frac12)\) \(\approx\) \(0.831940 - 0.623965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.37 + 1.25i)T \)
31 \( 1 + (-1.53 + 5.35i)T \)
good2 \( 1 + (0.626 + 2.33i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 1.39iT - 3T^{2} \)
5 \( 1 + (0.0315 + 0.117i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.74 - 0.468i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.33 - 0.358i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.20 + 5.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.21 - 8.25i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.44 + 5.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.28 + 1.31i)T + (14.5 + 25.1i)T^{2} \)
37 \( 1 + (1.23 - 1.23i)T - 37iT^{2} \)
41 \( 1 + (1.03 + 0.277i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.03 - 3.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.84 - 6.84i)T + 47iT^{2} \)
53 \( 1 + (-7.94 - 4.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.98 - 2.67i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.31 - 0.756i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.27 + 1.94i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.240 + 0.240i)T - 71iT^{2} \)
73 \( 1 + (2.29 + 8.55i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.9 + 8.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.47 - 0.931i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.64 - 9.86i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.61 + 1.77i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.80494322395506832146498808925, −10.26559710904284183771520392787, −9.524700470503204835320296596404, −8.947378597500130116875550149424, −7.73360942346364369211653217276, −6.14870876617870376041028834114, −4.69088032448520882116689178374, −3.70137301441265326731656132667, −2.90061539763043846890382382870, −1.13208419098048229884790369295, 1.15757325564494537338365448090, 3.56758629419989580408985467236, 5.10095316662244624093775719992, 6.16924274639953937611454897548, 6.94184868631871615967064052217, 7.34629670740382283410113312658, 8.584602009113344915414750378626, 9.151536731932086523066639859853, 10.23009602799644625626084622713, 11.43369278642766604761535225085

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