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Properties

Label 2-403-13.4-c1-0-3
Degree $2$
Conductor $403$
Sign $0.0449 - 0.998i$
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  + (−0.441 − 0.254i)2-s + (−1.38 + 2.39i)3-s + (−0.870 − 1.50i)4-s − 1.59i·5-s + (1.22 − 0.705i)6-s + (0.253 − 0.146i)7-s + 1.90i·8-s + (−2.33 − 4.04i)9-s + (−0.405 + 0.701i)10-s + (1.95 + 1.13i)11-s + 4.81·12-s + (3.01 + 1.98i)13-s − 0.149·14-s + (3.81 + 2.20i)15-s + (−1.25 + 2.17i)16-s + (1.26 + 2.19i)17-s + ⋯
L(s)  = 1  + (−0.311 − 0.180i)2-s + (−0.799 + 1.38i)3-s + (−0.435 − 0.753i)4-s − 0.711i·5-s + (0.498 − 0.287i)6-s + (0.0959 − 0.0553i)7-s + 0.673i·8-s + (−0.777 − 1.34i)9-s + (−0.128 + 0.221i)10-s + (0.590 + 0.341i)11-s + 1.39·12-s + (0.835 + 0.549i)13-s − 0.0399·14-s + (0.984 + 0.568i)15-s + (−0.313 + 0.543i)16-s + (0.307 + 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0449 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0449 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.0449 - 0.998i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.0449 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482191 + 0.460981i\)
\(L(\frac12)\) \(\approx\) \(0.482191 + 0.460981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.01 - 1.98i)T \)
31 \( 1 - iT \)
good2 \( 1 + (0.441 + 0.254i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.38 - 2.39i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.59iT - 5T^{2} \)
7 \( 1 + (-0.253 + 0.146i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.95 - 1.13i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.26 - 2.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.32 - 3.65i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.81 - 4.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.708 + 1.22i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (2.75 + 1.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.80 - 2.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.79 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.44iT - 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + (-4.18 + 2.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.654 + 1.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.75 - 1.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.23 + 0.714i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.52iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 16.5iT - 83T^{2} \)
89 \( 1 + (2.00 + 1.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.02 - 4.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.12112357900081154337243369567, −10.56973539797764337226717189195, −9.688265587400829206487592476682, −9.114380970939911878327715208035, −8.241416912354957860290003158211, −6.26126839050425938431294762245, −5.64754666613349609791105279766, −4.49115685315166429932233904756, −4.02709972986939067523618459067, −1.42693144687159663997425903527, 0.59504854831871053806061135468, 2.51082528241011034167908977306, 3.97714382473126800145585970738, 5.57674931927393573274808665779, 6.74324871684290404866550294651, 6.98868598957198710081380538202, 8.243371518832144461699853789235, 8.791028190499140119738242938372, 10.37209627754754267710746730723, 11.18399274731602964003623551231

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