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Properties

Label 2-403-13.12-c1-0-2
Degree $2$
Conductor $403$
Sign $0.977 - 0.209i$
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  − 2.71i·2-s + 0.265·3-s − 5.36·4-s + 0.954i·5-s − 0.721i·6-s + 4.28i·7-s + 9.13i·8-s − 2.92·9-s + 2.58·10-s + 0.0225i·11-s − 1.42·12-s + (0.755 + 3.52i)13-s + 11.6·14-s + 0.253i·15-s + 14.0·16-s − 4.73·17-s + ⋯
L(s)  = 1  − 1.91i·2-s + 0.153·3-s − 2.68·4-s + 0.426i·5-s − 0.294i·6-s + 1.61i·7-s + 3.22i·8-s − 0.976·9-s + 0.818·10-s + 0.00678i·11-s − 0.411·12-s + (0.209 + 0.977i)13-s + 3.10·14-s + 0.0654i·15-s + 3.51·16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653440 + 0.0692657i\)
\(L(\frac12)\) \(\approx\) \(0.653440 + 0.0692657i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.755 - 3.52i)T \)
31 \( 1 - iT \)
good2 \( 1 + 2.71iT - 2T^{2} \)
3 \( 1 - 0.265T + 3T^{2} \)
5 \( 1 - 0.954iT - 5T^{2} \)
7 \( 1 - 4.28iT - 7T^{2} \)
11 \( 1 - 0.0225iT - 11T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 + 1.51iT - 19T^{2} \)
23 \( 1 + 7.63T + 23T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
37 \( 1 + 2.46iT - 37T^{2} \)
41 \( 1 - 6.00iT - 41T^{2} \)
43 \( 1 - 9.18T + 43T^{2} \)
47 \( 1 + 6.05iT - 47T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
59 \( 1 - 8.26iT - 59T^{2} \)
61 \( 1 - 9.28T + 61T^{2} \)
67 \( 1 - 7.52iT - 67T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 - 1.76iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 - 6.66iT - 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−11.42096909230972130329776978513, −10.69663411392135735976762705023, −9.424755182912250136311848689395, −8.980549374664927104454213063919, −8.258935562039015828478262271708, −6.23844685099547457350647770252, −5.16396320088319407315040491361, −3.94112232960077448631019972841, −2.66480443416022957560624528638, −2.12962625862493964227764309698, 0.41489555450784126834565589529, 3.70528574463233296675654393930, 4.59342993213682063687058467321, 5.69224525312666656844107659800, 6.54064012908674438437497814498, 7.58498493506782658899945236248, 8.147965551622899697425297600728, 8.985865351683217631060460951519, 10.01682378628377072765707694943, 10.97948357577349487255105690701

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