Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.0137 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.169 + 1.60i)3-s + (0.309 − 0.951i)4-s + (−1.22 + 2.11i)5-s + (−0.809 − 1.40i)6-s + (0.809 − 0.171i)7-s + (0.309 + 0.951i)8-s + (0.373 + 0.0794i)9-s + (−0.255 − 2.43i)10-s + (−1.39 − 1.54i)11-s + (1.47 + 0.658i)12-s + (4.27 − 1.90i)13-s + (−0.553 + 0.614i)14-s + (−3.20 − 2.32i)15-s + (−0.809 − 0.587i)16-s + (4.03 − 4.47i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.0976 + 0.929i)3-s + (0.154 − 0.475i)4-s + (−0.546 + 0.946i)5-s + (−0.330 − 0.572i)6-s + (0.305 − 0.0649i)7-s + (0.109 + 0.336i)8-s + (0.124 + 0.0264i)9-s + (−0.0808 − 0.768i)10-s + (−0.420 − 0.467i)11-s + (0.426 + 0.189i)12-s + (1.18 − 0.527i)13-s + (−0.147 + 0.164i)14-s + (−0.826 − 0.600i)15-s + (−0.202 − 0.146i)16-s + (0.978 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0137 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0137 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.0137 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (59, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.0137 - 0.999i)$
$L(1)$  $\approx$  $0.470750 + 0.464337i$
$L(\frac12)$  $\approx$  $0.470750 + 0.464337i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-2.33 + 5.05i)T \)
good3 \( 1 + (0.169 - 1.60i)T + (-2.93 - 0.623i)T^{2} \)
5 \( 1 + (1.22 - 2.11i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.809 + 0.171i)T + (6.39 - 2.84i)T^{2} \)
11 \( 1 + (1.39 + 1.54i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-4.27 + 1.90i)T + (8.69 - 9.66i)T^{2} \)
17 \( 1 + (-4.03 + 4.47i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (5.75 + 2.56i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (-1.57 - 4.85i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.37 - 3.90i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (-1.98 - 3.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.10 + 10.4i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (3.70 + 1.64i)T + (28.7 + 31.9i)T^{2} \)
47 \( 1 + (3.74 + 2.71i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.59 + 0.339i)T + (48.4 + 21.5i)T^{2} \)
59 \( 1 + (-1.15 + 10.9i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + (-0.0971 + 0.168i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.12 - 1.08i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (0.985 + 1.09i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (8.33 - 9.26i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-0.0835 - 0.794i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + (0.991 - 3.05i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.98 - 12.2i)T + (-78.4 - 57.0i)T^{2} \)
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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

−15.45686847736836130194654256146, −14.68954390823774864839457645341, −13.30900580519782699027495932691, −11.25725293808583469020234349117, −10.79609040891653510164813643794, −9.620909318892001257341062802283, −8.211308926952794682552321166483, −7.01422586599828320761250609457, −5.35222001588186308591548519493, −3.52266908413777707220955037555, 1.53072920549421909342227750867, 4.23654806967432294815915939694, 6.37092260269743223941570125498, 7.942742103039210999001703192058, 8.553827133276685874675246127475, 10.21338404685466022907909396804, 11.53690937447842268367091552615, 12.63408609976675914964781882702, 13.01064868525051717876811631413, 14.77836192714674216072736830278

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