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Properties

Label 1-2668-2668.695-r0-0-0
Degree $1$
Conductor $2668$
Sign $0.987 + 0.159i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.142 − 0.989i)3-s + (0.959 + 0.281i)5-s + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.654 − 0.755i)13-s + (0.142 − 0.989i)15-s + (0.415 − 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (0.841 + 0.540i)25-s + (0.415 + 0.909i)27-s + (−0.142 + 0.989i)31-s + (0.654 + 0.755i)33-s + (−0.841 + 0.540i)35-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)3-s + (0.959 + 0.281i)5-s + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.654 − 0.755i)13-s + (0.142 − 0.989i)15-s + (0.415 − 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (0.841 + 0.540i)25-s + (0.415 + 0.909i)27-s + (−0.142 + 0.989i)31-s + (0.654 + 0.755i)33-s + (−0.841 + 0.540i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (695, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.263025933 + 0.1013936438i\)
\(L(\frac12)\) \(\approx\) \(1.263025933 + 0.1013936438i\)
\(L(1)\) \(\approx\) \(0.9602693224 - 0.1356640527i\)
\(L(1)\) \(\approx\) \(0.9602693224 - 0.1356640527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.20860139584938609636610869742, −18.76586893089807148500567644999, −17.51791019557911553455025673927, −17.044639896404626499603961204298, −16.571303805360728561605879701274, −15.95017025523714412323961972222, −15.07165534343209401780092076921, −14.174790709992083285631285704906, −13.84955259318648929182395484265, −12.84058426339757562799000657223, −12.294290961514089298803341906574, −11.11868180596016318059848585171, −10.401783930592846223152772051309, −10.07841610923301180290666402860, −9.3205516080667307338402372900, −8.617797001205337507237394120568, −7.68707223701356007260984180887, −6.634818538823744342204075482628, −5.81957194067650273371906943177, −5.394549822560854872852852070, −4.29362388492404518126472487813, −3.764908817050151437980427636146, −2.74953964063935951684280413005, −1.88953711863933958148025117396, −0.50917281759248765561320807392, 0.80193852778029911964299508035, 2.027143277096586756342059968566, 2.67506718897028728820168565780, 3.04157383373521964855215450313, 4.870302807896697082955065849089, 5.43549937209182214972208098549, 6.04991324729519026818584258816, 7.00508889468862874364064844336, 7.34897268380156366704634099347, 8.46997838822135439804308969044, 9.1720054336553890164007672744, 9.99227065387782053996130534234, 10.60667801542039108141545375111, 11.61124234543481340827358648682, 12.416826667360587199250051130127, 12.90819147348262626071960113370, 13.43733925673385260605540354069, 14.283376059323072812419948587380, 14.98578436723601652843967310509, 15.78972458786543779869439662111, 16.637389231582702633271577950370, 17.5323490371608022741895431943, 17.98241526703161424110807715544, 18.41427334257808806045575890163, 19.30382378661115937308448936900

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