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Properties

Label 2-31e2-31.2-c1-0-54
Degree $2$
Conductor $961$
Sign $-0.997 + 0.0753i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
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  + (−0.335 − 0.243i)2-s + (0.335 − 0.243i)3-s + (−0.565 − 1.73i)4-s + 5-s − 0.171·6-s + (−0.127 − 0.393i)7-s + (−0.490 + 1.50i)8-s + (−0.874 + 2.68i)9-s + (−0.335 − 0.243i)10-s + (−1.00 − 3.08i)11-s + (−0.612 − 0.445i)12-s + (−3.09 + 2.25i)13-s + (−0.0530 + 0.163i)14-s + (0.335 − 0.243i)15-s + (−2.42 + 1.76i)16-s + (1.80 − 5.54i)17-s + ⋯
L(s)  = 1  + (−0.236 − 0.172i)2-s + (0.193 − 0.140i)3-s + (−0.282 − 0.869i)4-s + 0.447·5-s − 0.0700·6-s + (−0.0483 − 0.148i)7-s + (−0.173 + 0.533i)8-s + (−0.291 + 0.896i)9-s + (−0.105 − 0.0769i)10-s + (−0.302 − 0.929i)11-s + (−0.176 − 0.128i)12-s + (−0.859 + 0.624i)13-s + (−0.0141 + 0.0436i)14-s + (0.0865 − 0.0628i)15-s + (−0.606 + 0.440i)16-s + (0.436 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.997 + 0.0753i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.997 + 0.0753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0233267 - 0.617873i\)
\(L(\frac12)\) \(\approx\) \(0.0233267 - 0.617873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.335 + 0.243i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.335 + 0.243i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (0.127 + 0.393i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.00 + 3.08i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.09 - 2.25i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.80 + 5.54i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.57 + 2.59i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.23 + 3.80i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.52 + 4.01i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-6.05 - 4.39i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (8.81 + 6.40i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (7.81 - 5.67i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.80 - 5.54i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.29 - 2.39i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 - 3.24T + 67T^{2} \)
71 \( 1 + (-0.0219 + 0.0675i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.565 - 1.73i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.08 + 6.42i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (8.14 + 5.91i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.38 + 4.26i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.59 - 4.91i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.624473897633268696458034876125, −8.955637560229606784277821570906, −8.077353363318389909777278706162, −7.10253411403187890065378400369, −6.05514154414205557248586571350, −5.25443760442636536626899706409, −4.50497547103155691378959651773, −2.77818011815767627582475851594, −1.95791126517290227916977239341, −0.28218391925381551714308506094, 1.97528934845772330902290957294, 3.26912413271597384975892766400, 4.02364988197992490867794547603, 5.25747623449343855498335853548, 6.22592049219369445286279461447, 7.22484788100807740332813307316, 7.997849639512335563190360846203, 8.714623845014306132443106274695, 9.721216737444878659724818637144, 9.963093475571762571167269443385

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