Properties

Label 2-403-403.25-c1-0-0
Degree $2$
Conductor $403$
Sign $0.476 - 0.879i$
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  − 2.42i·2-s + (1.37 + 2.37i)3-s − 3.89·4-s + (−2.09 − 1.20i)5-s + (5.77 − 3.33i)6-s + (−4.33 + 2.50i)7-s + 4.60i·8-s + (−2.26 + 3.92i)9-s + (−2.93 + 5.08i)10-s + (4.09 + 2.36i)11-s + (−5.34 − 9.26i)12-s + (−0.352 + 3.58i)13-s + (6.07 + 10.5i)14-s − 6.64i·15-s + 3.38·16-s + (−1.24 − 2.15i)17-s + ⋯
L(s)  = 1  − 1.71i·2-s + (0.792 + 1.37i)3-s − 1.94·4-s + (−0.936 − 0.540i)5-s + (2.35 − 1.36i)6-s + (−1.63 + 0.945i)7-s + 1.62i·8-s + (−0.756 + 1.30i)9-s + (−0.928 + 1.60i)10-s + (1.23 + 0.712i)11-s + (−1.54 − 2.67i)12-s + (−0.0978 + 0.995i)13-s + (1.62 + 2.81i)14-s − 1.71i·15-s + 0.846·16-s + (−0.302 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.476 - 0.879i$
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: \((2,\ 403,\ (\ :1/2),\ 0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591395 + 0.352125i\)
\(L(\frac12)\) \(\approx\) \(0.591395 + 0.352125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.352 - 3.58i)T \)
31 \( 1 + (1.51 - 5.35i)T \)
good2 \( 1 + 2.42iT - 2T^{2} \)
3 \( 1 + (-1.37 - 2.37i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.09 + 1.20i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (4.33 - 2.50i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.24 + 2.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.12 - 2.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.407T + 23T^{2} \)
29 \( 1 - 0.937T + 29T^{2} \)
37 \( 1 + (-2.00 + 1.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.68 + 5.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.428 - 0.741i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.40iT - 47T^{2} \)
53 \( 1 + (5.12 - 8.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.00 + 5.20i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.35T + 61T^{2} \)
67 \( 1 + (-7.21 - 4.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.24 + 1.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.38 - 3.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.94 + 6.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.55 - 2.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.15iT - 89T^{2} \)
97 \( 1 + 4.77iT - 97T^{2} \)
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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

−11.50911162122547173136963344045, −10.30313778778059724320911821391, −9.651614369666016102064076131520, −9.003366245056598086208724316638, −8.696537434653946575349920389286, −6.66789319151569364292679812935, −4.79926426403681272781951264468, −3.97627577003605985674999526382, −3.46372980899161854034350856992, −2.25698670003463587600476802416, 0.40223995330058259142104516384, 3.18381757938839384797098889116, 4.02346158955935101056528225303, 6.14289301431551871720429856174, 6.65371170008060624022996626486, 7.18412881803780074287589957756, 8.023494529989032570530581905182, 8.691875566840748727713710047842, 9.752757056802135814095738363201, 11.17480503935608899873284294350

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