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Properties

Label 2-403-13.10-c1-0-16
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: $\chi_{403} (218, \cdot )$
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

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

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