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Properties

Label 2-403-403.25-c1-0-31
Degree $2$
Conductor $403$
Sign $-0.821 - 0.570i$
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  − 1.25i·2-s + (−1.30 − 2.26i)3-s + 0.423·4-s + (−3.09 − 1.78i)5-s + (−2.83 + 1.63i)6-s + (4.10 − 2.36i)7-s − 3.04i·8-s + (−1.90 + 3.30i)9-s + (−2.24 + 3.88i)10-s + (−2.82 − 1.63i)11-s + (−0.552 − 0.956i)12-s + (3.60 − 0.135i)13-s + (−2.97 − 5.15i)14-s + 9.33i·15-s − 2.97·16-s + (3.60 + 6.24i)17-s + ⋯
L(s)  = 1  − 0.887i·2-s + (−0.753 − 1.30i)3-s + 0.211·4-s + (−1.38 − 0.799i)5-s + (−1.15 + 0.669i)6-s + (1.55 − 0.895i)7-s − 1.07i·8-s + (−0.635 + 1.10i)9-s + (−0.709 + 1.22i)10-s + (−0.852 − 0.492i)11-s + (−0.159 − 0.276i)12-s + (0.999 − 0.0374i)13-s + (−0.795 − 1.37i)14-s + 2.41i·15-s − 0.743·16-s + (0.874 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.821 - 0.570i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.821 - 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321699 + 1.02714i\)
\(L(\frac12)\) \(\approx\) \(0.321699 + 1.02714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.60 + 0.135i)T \)
31 \( 1 + (2.77 - 4.82i)T \)
good2 \( 1 + 1.25iT - 2T^{2} \)
3 \( 1 + (1.30 + 2.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.09 + 1.78i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-4.10 + 2.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.82 + 1.63i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.60 - 6.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.759 + 0.438i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 - 8.05T + 29T^{2} \)
37 \( 1 + (0.365 - 0.210i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.53 - 2.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.23 - 2.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.10iT - 47T^{2} \)
53 \( 1 + (-2.26 + 3.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.16 + 2.40i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 + (7.22 + 4.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.82 + 5.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.353 - 0.203i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.65 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.29 - 5.36i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 6.98iT - 97T^{2} \)
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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

−10.88981666397377848891653719031, −10.60787241597400861384557991937, −8.349619085724302158762082253013, −7.991711677328313934030778927794, −7.20520019422958054695078121493, −5.94516975338794898874239876256, −4.63971885176399752284527858873, −3.56143693798708909542923095762, −1.53380165768927031806243288854, −0.867208902451422026354277284097, 2.77827792802436796890663113337, 4.29818964473113951878989189881, 5.13325484438055095865969161246, 5.86084348440121107139204539459, 7.33217614563831983945858899927, 7.894225854445312378082902981262, 8.809535688342096677770513298508, 10.31526753254651531167218241278, 11.01633051535137618424511113365, 11.66281285229673985204549229237

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