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Properties

Label 2-403-403.25-c1-0-29
Degree $2$
Conductor $403$
Sign $-0.985 - 0.172i$
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  − 1.84i·2-s + (−1.26 − 2.18i)3-s − 1.40·4-s + (3.26 + 1.88i)5-s + (−4.03 + 2.32i)6-s + (1.26 − 0.729i)7-s − 1.10i·8-s + (−1.68 + 2.91i)9-s + (3.47 − 6.01i)10-s + (−5.25 − 3.03i)11-s + (1.76 + 3.06i)12-s + (−3.59 − 0.302i)13-s + (−1.34 − 2.33i)14-s − 9.51i·15-s − 4.83·16-s + (−0.00325 − 0.00563i)17-s + ⋯
L(s)  = 1  − 1.30i·2-s + (−0.728 − 1.26i)3-s − 0.701·4-s + (1.45 + 0.842i)5-s + (−1.64 + 0.950i)6-s + (0.477 − 0.275i)7-s − 0.389i·8-s + (−0.561 + 0.973i)9-s + (1.09 − 1.90i)10-s + (−1.58 − 0.914i)11-s + (0.510 + 0.884i)12-s + (−0.996 − 0.0837i)13-s + (−0.359 − 0.623i)14-s − 2.45i·15-s − 1.20·16-s + (−0.000789 − 0.00136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.985 - 0.172i$
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: \((2,\ 403,\ (\ :1/2),\ -0.985 - 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109782 + 1.26324i\)
\(L(\frac12)\) \(\approx\) \(0.109782 + 1.26324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.59 + 0.302i)T \)
31 \( 1 + (0.183 + 5.56i)T \)
good2 \( 1 + 1.84iT - 2T^{2} \)
3 \( 1 + (1.26 + 2.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.26 - 1.88i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.26 + 0.729i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.25 + 3.03i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.00325 + 0.00563i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.85 + 2.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.95T + 23T^{2} \)
29 \( 1 - 0.860T + 29T^{2} \)
37 \( 1 + (-5.71 + 3.30i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.392 + 0.226i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.15 - 3.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.08iT - 47T^{2} \)
53 \( 1 + (0.367 - 0.637i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.33 + 4.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 9.39T + 61T^{2} \)
67 \( 1 + (-12.5 - 7.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.6 + 6.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.05 - 2.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.85 - 4.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.45 - 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.26iT - 89T^{2} \)
97 \( 1 + 5.36iT - 97T^{2} \)
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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

−11.15797970522356921616615511606, −10.18980019553974615119528744707, −9.451926311138260087468899021538, −7.75105230191683608418899381968, −6.98681130900689843197878000925, −5.95668454716261770071667469661, −5.10117435210428923283286196658, −2.84886404121483773962515139542, −2.29258285656333507465071535531, −0.905576090883992837562134951996, 2.30833059331539785731405612682, 4.79819478993698416114763080363, 5.18626951325353727175361460957, 5.54515027924154129561991390616, 6.90051165651733884998387958643, 8.026918838925697415521586892482, 9.088151542891913236425378239120, 9.890325333553758599247920617929, 10.42191189040151533470003079386, 11.65642981674774448290575439086

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