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Properties

Label 2-403-403.36-c1-0-20
Degree $2$
Conductor $403$
Sign $0.628 + 0.777i$
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  + (−0.793 + 0.457i)2-s − 1.36·3-s + (−0.580 + 1.00i)4-s + (3.20 − 1.84i)5-s + (1.08 − 0.627i)6-s + (1.32 − 0.762i)7-s − 2.89i·8-s − 1.12·9-s + (−1.69 + 2.93i)10-s + (−2.03 − 1.17i)11-s + (0.795 − 1.37i)12-s + (−1.76 + 3.14i)13-s + (−0.698 + 1.20i)14-s + (−4.38 + 2.53i)15-s + (0.165 + 0.285i)16-s + (−3.85 − 6.66i)17-s + ⋯
L(s)  = 1  + (−0.560 + 0.323i)2-s − 0.790·3-s + (−0.290 + 0.502i)4-s + (1.43 − 0.826i)5-s + (0.443 − 0.256i)6-s + (0.499 − 0.288i)7-s − 1.02i·8-s − 0.374·9-s + (−0.535 + 0.927i)10-s + (−0.614 − 0.354i)11-s + (0.229 − 0.397i)12-s + (−0.489 + 0.871i)13-s + (−0.186 + 0.323i)14-s + (−1.13 + 0.653i)15-s + (0.0412 + 0.0714i)16-s + (−0.933 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.628 + 0.777i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689709 - 0.329164i\)
\(L(\frac12)\) \(\approx\) \(0.689709 - 0.329164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.76 - 3.14i)T \)
31 \( 1 + (-5.56 + 0.195i)T \)
good2 \( 1 + (0.793 - 0.457i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 1.36T + 3T^{2} \)
5 \( 1 + (-3.20 + 1.84i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.32 + 0.762i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.85 + 6.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.76 + 3.90i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.106 - 0.185i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.38 + 4.12i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + 2.48iT - 37T^{2} \)
41 \( 1 + (-1.08 - 0.626i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.06 - 1.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.38iT - 47T^{2} \)
53 \( 1 + (6.75 + 11.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.62 + 5.55i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.06 - 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.98 + 1.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.58iT - 71T^{2} \)
73 \( 1 + (4.64 - 2.68i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.55 - 2.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.57 + 4.37i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.51 + 2.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.78 + 1.03i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27242754528179476344865723603, −9.836588885324612446531677329249, −9.376234924526001913520993908664, −8.554652726600767337561833214146, −7.38806064525883092930232727474, −6.43731590489628937620347320935, −5.20577109318239086961235046434, −4.73078314250062480480618589406, −2.60935397164782136587028531764, −0.68566608672144086140795321246, 1.60188994568109170083628568228, 2.75294126825476895874159114607, 5.03818581983971107359899810231, 5.65559401487723736140125612829, 6.31516470291772489256886350811, 7.81257891700179593958738826228, 8.879236863651082855975042154875, 9.939806393810262404296521404275, 10.44790124712846034492181665748, 10.96764275380945289178334832919

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