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Properties

Label 2-403-31.16-c1-0-5
Degree $2$
Conductor $403$
Sign $0.463 - 0.886i$
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  + (1.10 − 0.805i)2-s + (−1.36 − 0.993i)3-s + (−0.0371 + 0.114i)4-s − 0.800·5-s − 2.31·6-s + (−1.47 + 4.54i)7-s + (0.898 + 2.76i)8-s + (−0.0450 − 0.138i)9-s + (−0.887 + 0.645i)10-s + (−0.368 + 1.13i)11-s + (0.164 − 0.119i)12-s + (0.809 + 0.587i)13-s + (2.02 + 6.22i)14-s + (1.09 + 0.794i)15-s + (3.03 + 2.20i)16-s + (0.638 + 1.96i)17-s + ⋯
L(s)  = 1  + (0.784 − 0.569i)2-s + (−0.789 − 0.573i)3-s + (−0.0185 + 0.0571i)4-s − 0.357·5-s − 0.945·6-s + (−0.557 + 1.71i)7-s + (0.317 + 0.977i)8-s + (−0.0150 − 0.0462i)9-s + (−0.280 + 0.203i)10-s + (−0.111 + 0.342i)11-s + (0.0473 − 0.0344i)12-s + (0.224 + 0.163i)13-s + (0.540 + 1.66i)14-s + (0.282 + 0.205i)15-s + (0.757 + 0.550i)16-s + (0.154 + 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.463 - 0.886i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.463 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886009 + 0.536468i\)
\(L(\frac12)\) \(\approx\) \(0.886009 + 0.536468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-2.65 + 4.89i)T \)
good2 \( 1 + (-1.10 + 0.805i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.36 + 0.993i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 0.800T + 5T^{2} \)
7 \( 1 + (1.47 - 4.54i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.368 - 1.13i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (-0.638 - 1.96i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.05 - 1.49i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.203 - 0.625i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.84 + 1.34i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 6.15T + 37T^{2} \)
41 \( 1 + (-1.08 + 0.787i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.02 - 5.10i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-2.38 - 1.73i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.82 - 11.7i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.60 + 2.61i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 5.00T + 61T^{2} \)
67 \( 1 + 2.60T + 67T^{2} \)
71 \( 1 + (-3.13 - 9.65i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.64 + 14.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.46 - 4.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-10.2 + 7.46i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.50 + 4.63i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.345 + 1.06i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.79578947129971337354089033597, −11.05610241341829023732420643674, −9.658626274900634714692930669973, −8.637818914311233100512226137356, −7.71257246455794119984620180891, −6.19511039891071711532402895432, −5.79540630760736548349508968424, −4.52305430779626129388950476165, −3.25129273362840546256230922522, −2.08317343806396547487346648423, 0.57738369207005665251701728248, 3.53511777309932375721968391675, 4.36477697025811354711314617280, 5.16075330625907451306386423304, 6.27666959159652164513837901422, 7.00867225177514213099456925986, 8.025325300632780688016062709782, 9.647134850415886053826335561465, 10.39846611329503748919902877385, 10.89350536628900384634413218880

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