Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.684 - 0.729i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.35·2-s + (1.34 + 2.33i)3-s + 3.52·4-s + (−0.0458 + 0.0794i)5-s + (3.16 + 5.48i)6-s + (−2.00 − 3.47i)7-s + 3.59·8-s + (−2.13 + 3.69i)9-s + (−0.107 + 0.186i)10-s + (0.00492 − 0.00853i)11-s + (4.75 + 8.23i)12-s + (−0.5 + 0.866i)13-s + (−4.71 − 8.15i)14-s − 0.247·15-s + 1.39·16-s + (−1.21 − 2.10i)17-s + ⋯
L(s)  = 1  + 1.66·2-s + (0.777 + 1.34i)3-s + 1.76·4-s + (−0.0205 + 0.0355i)5-s + (1.29 + 2.24i)6-s + (−0.757 − 1.31i)7-s + 1.27·8-s + (−0.710 + 1.23i)9-s + (−0.0341 + 0.0591i)10-s + (0.00148 − 0.00257i)11-s + (1.37 + 2.37i)12-s + (−0.138 + 0.240i)13-s + (−1.25 − 2.18i)14-s − 0.0638·15-s + 0.348·16-s + (−0.294 − 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.684 - 0.729i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.684 - 0.729i)\)
\(L(1)\)  \(\approx\)  \(3.44495 + 1.49140i\)
\(L(\frac12)\)  \(\approx\)  \(3.44495 + 1.49140i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.597 - 5.53i)T \)
good2 \( 1 - 2.35T + 2T^{2} \)
3 \( 1 + (-1.34 - 2.33i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.0458 - 0.0794i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.00 + 3.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.00492 + 0.00853i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.21 + 2.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 + 4.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.809T + 23T^{2} \)
29 \( 1 - 2.84T + 29T^{2} \)
37 \( 1 + (2.80 + 4.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.56 - 4.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.90 - 10.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 + (-1.42 + 2.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.22 + 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (2.78 - 4.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.09 - 1.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.50 + 2.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.54 + 2.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.24 + 7.35i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.36T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.25566041775326834230445968556, −10.66665233142246620917157820454, −9.722935932259023658861835321661, −8.852041691507047187446727412186, −7.29250561759332595855860219163, −6.51163181138190933014516110179, −5.03001419385983586456463327245, −4.35800275497861708022134812983, −3.56500714615527623122176058725, −2.76702421621569474266746184672, 2.14496980962208518462283533315, 2.79119821504423339515987401359, 3.99452707367104707154004086011, 5.58145447848288044741182687883, 6.23379039795754896955788204222, 7.02411841196893059972763957501, 8.233948355251658753430957722382, 9.004667439277094850770876263428, 10.49362791417602021006797093363, 12.05377889306968619891503542352

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