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Properties

Label 2-678-113.102-c1-0-18
Degree $2$
Conductor $678$
Sign $-0.698 - 0.715i$
Analytic cond. $5.41385$
Root an. cond. $2.32676$
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  + (−0.974 + 0.222i)2-s + (−0.875 − 0.483i)3-s + (0.900 − 0.433i)4-s + (−1.96 − 3.56i)5-s + (0.960 + 0.276i)6-s + (−1.67 − 2.10i)7-s + (−0.781 + 0.623i)8-s + (0.532 + 0.846i)9-s + (2.71 + 3.03i)10-s + (−1.92 + 0.673i)11-s + (−0.998 − 0.0560i)12-s + (0.757 − 6.72i)13-s + (2.10 + 1.67i)14-s + 4.07i·15-s + (0.623 − 0.781i)16-s + (−4.80 + 3.41i)17-s + ⋯
L(s)  = 1  + (−0.689 + 0.157i)2-s + (−0.505 − 0.279i)3-s + (0.450 − 0.216i)4-s + (−0.880 − 1.59i)5-s + (0.392 + 0.113i)6-s + (−0.633 − 0.793i)7-s + (−0.276 + 0.220i)8-s + (0.177 + 0.282i)9-s + (0.857 + 0.959i)10-s + (−0.580 + 0.203i)11-s + (−0.288 − 0.0161i)12-s + (0.210 − 1.86i)13-s + (0.561 + 0.447i)14-s + 1.05i·15-s + (0.155 − 0.195i)16-s + (−1.16 + 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(678\)    =    \(2 \cdot 3 \cdot 113\)
Sign: $-0.698 - 0.715i$
Analytic conductor: \(5.41385\)
Root analytic conductor: \(2.32676\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{678} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 678,\ (\ :1/2),\ -0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0964575 + 0.229154i\)
\(L(\frac12)\) \(\approx\) \(0.0964575 + 0.229154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 - 0.222i)T \)
3 \( 1 + (0.875 + 0.483i)T \)
113 \( 1 + (-0.339 - 10.6i)T \)
good5 \( 1 + (1.96 + 3.56i)T + (-2.66 + 4.23i)T^{2} \)
7 \( 1 + (1.67 + 2.10i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (1.92 - 0.673i)T + (8.60 - 6.85i)T^{2} \)
13 \( 1 + (-0.757 + 6.72i)T + (-12.6 - 2.89i)T^{2} \)
17 \( 1 + (4.80 - 3.41i)T + (5.61 - 16.0i)T^{2} \)
19 \( 1 + (-3.38 + 0.976i)T + (16.0 - 10.1i)T^{2} \)
23 \( 1 + (-1.45 + 5.04i)T + (-19.4 - 12.2i)T^{2} \)
29 \( 1 + (-5.36 - 7.55i)T + (-9.57 + 27.3i)T^{2} \)
31 \( 1 + (-7.16 - 0.807i)T + (30.2 + 6.89i)T^{2} \)
37 \( 1 + (7.22 - 6.45i)T + (4.14 - 36.7i)T^{2} \)
41 \( 1 + (4.32 + 1.51i)T + (32.0 + 25.5i)T^{2} \)
43 \( 1 + (-0.825 + 4.85i)T + (-40.5 - 14.2i)T^{2} \)
47 \( 1 + (2.91 - 0.163i)T + (46.7 - 5.26i)T^{2} \)
53 \( 1 + (-1.16 - 2.42i)T + (-33.0 + 41.4i)T^{2} \)
59 \( 1 + (0.289 + 1.00i)T + (-49.9 + 31.3i)T^{2} \)
61 \( 1 + (-0.0374 - 0.106i)T + (-47.6 + 38.0i)T^{2} \)
67 \( 1 + (0.397 - 7.08i)T + (-66.5 - 7.50i)T^{2} \)
71 \( 1 + (6.42 + 15.5i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-1.53 - 0.637i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (10.2 - 11.4i)T + (-8.84 - 78.5i)T^{2} \)
83 \( 1 + (1.75 + 7.66i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (8.43 - 1.43i)T + (84.0 - 29.3i)T^{2} \)
97 \( 1 + (-2.44 + 3.06i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.24659745305910090660078937036, −8.705894531516606127503751287882, −8.381286907832773648394021095139, −7.46171592589679710096082712167, −6.58045389761972563152312724744, −5.32372348806053708022002566292, −4.61203914820298280566385162402, −3.21093420032377441118400915942, −1.12150052777304351625316124886, −0.20453340852551125086694425372, 2.38611703211259008174048493026, 3.27708929839258694399990738154, 4.44630088642847483297732655961, 6.04724680186877601019221614896, 6.74959122692336270591026935020, 7.36272081106828626598354472012, 8.540313822414030382790678702418, 9.506196673922443223192813481736, 10.13626590561151797088666579951, 11.21402438872195248579777247839

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