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Properties

Label 2-671-61.34-c1-0-35
Degree $2$
Conductor $671$
Sign $0.825 + 0.564i$
Analytic cond. $5.35796$
Root an. cond. $2.31472$
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  + (−0.200 + 0.145i)2-s + (−0.573 + 0.416i)3-s + (−0.599 + 1.84i)4-s + (1.29 − 3.99i)5-s + (0.0542 − 0.166i)6-s + (−0.123 − 0.0894i)7-s + (−0.301 − 0.927i)8-s + (−0.771 + 2.37i)9-s + (0.321 + 0.989i)10-s − 11-s + (−0.424 − 1.30i)12-s + 3.24·13-s + 0.0376·14-s + (0.920 + 2.83i)15-s + (−2.94 − 2.13i)16-s + (1.30 − 4.01i)17-s + ⋯
L(s)  = 1  + (−0.141 + 0.102i)2-s + (−0.331 + 0.240i)3-s + (−0.299 + 0.921i)4-s + (0.580 − 1.78i)5-s + (0.0221 − 0.0681i)6-s + (−0.0465 − 0.0338i)7-s + (−0.106 − 0.327i)8-s + (−0.257 + 0.791i)9-s + (0.101 + 0.312i)10-s − 0.301·11-s + (−0.122 − 0.377i)12-s + 0.900·13-s + 0.0100·14-s + (0.237 + 0.731i)15-s + (−0.735 − 0.534i)16-s + (0.316 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(5.35796\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 671,\ (\ :1/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14342 - 0.353588i\)
\(L(\frac12)\) \(\approx\) \(1.14342 - 0.353588i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
61 \( 1 + (-1.97 - 7.55i)T \)
good2 \( 1 + (0.200 - 0.145i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.573 - 0.416i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.29 + 3.99i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.123 + 0.0894i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + (-1.30 + 4.01i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.20 + 2.33i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.14 + 3.52i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 - 1.10T + 29T^{2} \)
31 \( 1 + (-6.04 - 4.39i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.77 + 2.01i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.63 - 4.81i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.48 + 7.63i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + (1.36 + 4.20i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.6 + 7.71i)T + (18.2 - 56.1i)T^{2} \)
67 \( 1 + (3.20 - 9.86i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.61 - 8.04i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.83 + 14.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.41 - 10.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (9.59 - 6.97i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.52 + 1.83i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.05 + 2.22i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.24671192296404977825923027026, −9.382070612430321218753402650711, −8.593393988331028303088604061028, −8.188989425560144243684173431631, −7.00376299492815156687533067250, −5.55977277057565439648230758173, −4.99605590922521849733022746296, −4.14020942950651036979738423830, −2.59979251798203906815175737124, −0.799800273302935664045435325358, 1.36550517044382405859095844751, 2.78771937940611802571782897078, 3.85745162608770339066099963886, 5.70854672397740333671251178434, 6.00418713452834345965346884873, 6.77475791405840880387016716288, 7.87015521123943255821622605312, 9.175076906561983526391403482691, 9.895404657590146901835468975900, 10.60911045744031895673245276228

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