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Properties

Label 2-670-335.104-c1-0-1
Degree $2$
Conductor $670$
Sign $-0.998 + 0.0469i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
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  + (−0.866 + 0.5i)2-s + 1.19i·3-s + (0.499 − 0.866i)4-s + (−1.06 − 1.96i)5-s + (−0.596 − 1.03i)6-s + (−0.996 − 0.575i)7-s + 0.999i·8-s + 1.57·9-s + (1.90 + 1.16i)10-s + (−0.209 + 0.363i)11-s + (1.03 + 0.596i)12-s + (−5.35 + 3.08i)13-s + 1.15·14-s + (2.34 − 1.27i)15-s + (−0.5 − 0.866i)16-s + (0.541 − 0.312i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 0.688i·3-s + (0.249 − 0.433i)4-s + (−0.478 − 0.878i)5-s + (−0.243 − 0.421i)6-s + (−0.376 − 0.217i)7-s + 0.353i·8-s + 0.525·9-s + (0.603 + 0.368i)10-s + (−0.0632 + 0.109i)11-s + (0.298 + 0.172i)12-s + (−1.48 + 0.856i)13-s + 0.307·14-s + (0.604 − 0.329i)15-s + (−0.125 − 0.216i)16-s + (0.131 − 0.0757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.998 + 0.0469i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.998 + 0.0469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00567917 - 0.241912i\)
\(L(\frac12)\) \(\approx\) \(0.00567917 - 0.241912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.06 + 1.96i)T \)
67 \( 1 + (-4.31 + 6.95i)T \)
good3 \( 1 - 1.19iT - 3T^{2} \)
7 \( 1 + (0.996 + 0.575i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.209 - 0.363i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.35 - 3.08i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.541 + 0.312i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.436 + 0.755i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.29 - 2.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.771 - 1.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.03 - 3.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.2 - 5.93i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.81 - 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 + (-6.63 - 3.83i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.04iT - 53T^{2} \)
59 \( 1 + 3.52T + 59T^{2} \)
61 \( 1 + (2.12 + 3.67i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (3.19 - 5.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (12.1 - 6.99i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.454 - 0.786i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.28 + 4.78i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.18T + 89T^{2} \)
97 \( 1 + (2.26 - 1.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.60540156018682634990949140375, −9.869966285396543802456933235648, −9.335289474331697538749376787716, −8.498753953884521798364468082381, −7.41587173810138099827948405521, −6.84888997359804557933766842826, −5.31431337492441602288728869078, −4.65262877949400599834256314157, −3.61616297677918652588916692890, −1.76416962521927650815758730459, 0.14991627893264447486825315790, 2.07542684380181192458316644160, 3.00602962873370060185062047544, 4.23106197324451798567447524261, 5.81241682328643029232524258059, 6.84740052523216389521078566152, 7.52690450370704598130816269770, 8.063914088338602881529365345972, 9.363625798038018330166245023535, 10.21520706812821722836015282664

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