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Properties

Label 2-670-67.29-c1-0-0
Degree $2$
Conductor $670$
Sign $0.437 - 0.899i$
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  + (0.5 − 0.866i)2-s − 3.12·3-s + (−0.499 − 0.866i)4-s − 5-s + (−1.56 + 2.70i)6-s + (−2.17 − 3.76i)7-s − 0.999·8-s + 6.74·9-s + (−0.5 + 0.866i)10-s + (−2.37 − 4.11i)11-s + (1.56 + 2.70i)12-s + (−2.63 + 4.56i)13-s − 4.34·14-s + 3.12·15-s + (−0.5 + 0.866i)16-s + (2.77 − 4.79i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 1.80·3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.637 + 1.10i)6-s + (−0.820 − 1.42i)7-s − 0.353·8-s + 2.24·9-s + (−0.158 + 0.273i)10-s + (−0.715 − 1.23i)11-s + (0.450 + 0.780i)12-s + (−0.731 + 1.26i)13-s − 1.16·14-s + 0.806·15-s + (−0.125 + 0.216i)16-s + (0.672 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0690278 + 0.0431838i\)
\(L(\frac12)\) \(\approx\) \(0.0690278 + 0.0431838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
67 \( 1 + (7.69 + 2.80i)T \)
good3 \( 1 + 3.12T + 3T^{2} \)
7 \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.37 + 4.11i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.63 - 4.56i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.77 + 4.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.45 - 2.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.25 - 2.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.83 - 3.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.72 - 4.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.22 + 3.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.01 - 5.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + (-3.50 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 + (-1.41 + 2.44i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (4.58 + 7.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.01 + 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.23 + 2.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.15 - 8.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + (3.60 - 6.24i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.85760089363123664329125056971, −10.17064872591815003023448127777, −9.461699213397199343086966149611, −7.71299114505889235407799163846, −6.87469759730366282649209545304, −6.13579258452698379991751904983, −5.06465720296299669777273823420, −4.31672261591865621671035321895, −3.25635227133659991454774970989, −1.02828588798501893625185905693, 0.06018456616055945462871966887, 2.59160599343982783125692471438, 4.25908717228026258646547650030, 5.21327438220569611393713934078, 5.75463282433648536524814096219, 6.52471175499019322357006010084, 7.45096656528773120509599008109, 8.381007082435419543929471087119, 9.901102149484990251834777067656, 10.19942413490485034204936391336

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