↑

Properties

Label 2-403-13.9-c1-0-22
Degree $2$
Conductor $403$
Sign $0.993 + 0.114i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.105 − 0.182i)2-s + (0.0693 − 0.120i)3-s + (0.977 + 1.69i)4-s + 1.89·5-s + (−0.0145 − 0.0252i)6-s + (−2.28 − 3.96i)7-s + 0.831·8-s + (1.49 + 2.58i)9-s + (0.199 − 0.345i)10-s + (1.94 − 3.36i)11-s + 0.271·12-s + (1.30 − 3.36i)13-s − 0.961·14-s + (0.131 − 0.227i)15-s + (−1.86 + 3.23i)16-s + (1.13 + 1.96i)17-s + ⋯
L(s)  = 1  + (0.0743 − 0.128i)2-s + (0.0400 − 0.0693i)3-s + (0.488 + 0.846i)4-s + 0.847·5-s + (−0.00595 − 0.0103i)6-s + (−0.864 − 1.49i)7-s + 0.294·8-s + (0.496 + 0.860i)9-s + (0.0630 − 0.109i)10-s + (0.585 − 1.01i)11-s + 0.0782·12-s + (0.361 − 0.932i)13-s − 0.257·14-s + (0.0339 − 0.0587i)15-s + (−0.467 + 0.808i)16-s + (0.275 + 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.993 + 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80884 - 0.103909i\)
\(L(\frac12)\) \(\approx\) \(1.80884 - 0.103909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.30 + 3.36i)T \)
31 \( 1 - T \)
good2 \( 1 + (-0.105 + 0.182i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0693 + 0.120i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.89T + 5T^{2} \)
7 \( 1 + (2.28 + 3.96i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.94 + 3.36i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.95 - 5.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.137 - 0.238i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.91 - 5.04i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-1.27 + 2.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.712 + 1.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.51 + 6.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.28T + 47T^{2} \)
53 \( 1 - 2.00T + 53T^{2} \)
59 \( 1 + (6.77 + 11.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.46 + 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.79 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.60 + 7.97i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.73T + 73T^{2} \)
79 \( 1 - 5.84T + 79T^{2} \)
83 \( 1 + 18.1T + 83T^{2} \)
89 \( 1 + (8.03 - 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.24 - 9.08i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.98124628032456428720703260427, −10.47538592093384944754780785670, −9.667162143114847740086895671120, −8.261640928130416300565615027746, −7.53229743735593190804166870848, −6.60713788821858498150085855410, −5.62913224692087259604091545263, −3.90086708929908736178043765515, −3.25079961153685881237109123904, −1.50915935222316016931734371946, 1.66577485020317792766153755407, 2.79967701266623676227592750124, 4.58029197667842475314803059583, 5.79670196201978836180834613688, 6.38872043920752890850618312308, 7.12618511920290420945610788877, 9.080995878354233920583950751300, 9.528949381313703898382694160405, 9.922118613511060237411453177719, 11.52781392056078183243991977784

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