↑

Properties

Label 2-403-403.123-c1-0-1
Degree $2$
Conductor $403$
Sign $0.916 - 0.400i$
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  + (−0.359 − 1.34i)2-s + (−2.72 − 1.57i)3-s + (0.0598 − 0.0345i)4-s + (1.15 − 1.15i)5-s + (−1.13 + 4.22i)6-s + (−1.21 + 4.51i)7-s + (−2.03 − 2.03i)8-s + (3.44 + 5.97i)9-s + (−1.96 − 1.13i)10-s + (1.19 + 4.46i)11-s − 0.217·12-s + (−2.59 + 2.50i)13-s + 6.50·14-s + (−4.96 + 1.32i)15-s + (−1.92 + 3.34i)16-s + (−1.22 − 2.12i)17-s + ⋯
L(s)  = 1  + (−0.254 − 0.949i)2-s + (−1.57 − 0.908i)3-s + (0.0299 − 0.0172i)4-s + (0.516 − 0.516i)5-s + (−0.461 + 1.72i)6-s + (−0.457 + 1.70i)7-s + (−0.718 − 0.718i)8-s + (1.14 + 1.99i)9-s + (−0.621 − 0.358i)10-s + (0.360 + 1.34i)11-s − 0.0628·12-s + (−0.719 + 0.694i)13-s + 1.73·14-s + (−1.28 + 0.343i)15-s + (−0.482 + 0.835i)16-s + (−0.297 − 0.516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376279 + 0.0786868i\)
\(L(\frac12)\) \(\approx\) \(0.376279 + 0.0786868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.59 - 2.50i)T \)
31 \( 1 + (5.28 + 1.73i)T \)
good2 \( 1 + (0.359 + 1.34i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (2.72 + 1.57i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.15 + 1.15i)T - 5iT^{2} \)
7 \( 1 + (1.21 - 4.51i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.19 - 4.46i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.15 + 0.846i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.22 - 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.31 - 2.48i)T + (14.5 + 25.1i)T^{2} \)
37 \( 1 + (1.09 + 4.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.673 + 2.51i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.14 - 7.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 - 1.79iT - 53T^{2} \)
59 \( 1 + (3.58 - 13.3i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.02 - 0.590i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.71 + 13.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.723 + 0.193i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.08 - 4.08i)T - 73iT^{2} \)
79 \( 1 - 1.76iT - 79T^{2} \)
83 \( 1 + (-8.36 - 8.36i)T + 83iT^{2} \)
89 \( 1 + (2.51 + 9.39i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.29 - 1.68i)T + (84.0 + 48.5i)T^{2} \)
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

−11.64704948284662863988967368078, −10.64614352032629124137171203555, −9.557313135099837960464427451657, −9.109868014569131942616923685567, −7.23731682872969328147808556514, −6.45690432925271202105834029696, −5.66743647845851468029345713346, −4.74528126960064281362276833807, −2.32895574937713837048312927679, −1.68617569852225452037586499496, 0.31438855728909243863763927814, 3.38325493037141323429224475839, 4.55003638847840706547669598888, 5.83511204280177866097893381445, 6.38114343748942721100322368456, 7.01582550085502148747745451537, 8.282840907827492594996029620833, 9.672057454535837090522443307527, 10.56986552903157076170737612868, 10.76660455850023591140959700344

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