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Properties

Label 1-3968-3968.1195-r0-0-0
Degree $1$
Conductor $3968$
Sign $-0.999 - 0.0428i$
Analytic cond. $18.4273$
Root an. cond. $18.4273$
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  + (−0.955 + 0.296i)3-s + (−0.896 − 0.442i)5-s + (0.824 + 0.566i)7-s + (0.824 − 0.566i)9-s + (0.801 − 0.598i)11-s + (0.533 + 0.845i)13-s + (0.987 + 0.156i)15-s + (0.998 + 0.0523i)17-s + (−0.169 + 0.985i)19-s + (−0.955 − 0.296i)21-s + (−0.996 + 0.0784i)23-s + (0.608 + 0.793i)25-s + (−0.619 + 0.785i)27-s + (−0.908 − 0.418i)29-s + (−0.587 + 0.809i)33-s + ⋯
L(s)  = 1  + (−0.955 + 0.296i)3-s + (−0.896 − 0.442i)5-s + (0.824 + 0.566i)7-s + (0.824 − 0.566i)9-s + (0.801 − 0.598i)11-s + (0.533 + 0.845i)13-s + (0.987 + 0.156i)15-s + (0.998 + 0.0523i)17-s + (−0.169 + 0.985i)19-s + (−0.955 − 0.296i)21-s + (−0.996 + 0.0784i)23-s + (0.608 + 0.793i)25-s + (−0.619 + 0.785i)27-s + (−0.908 − 0.418i)29-s + (−0.587 + 0.809i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $-0.999 - 0.0428i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ -0.999 - 0.0428i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005065006656 + 0.2361574229i\)
\(L(\frac12)\) \(\approx\) \(0.005065006656 + 0.2361574229i\)
\(L(1)\) \(\approx\) \(0.6767367162 + 0.1055600110i\)
\(L(1)\) \(\approx\) \(0.6767367162 + 0.1055600110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.955 + 0.296i)T \)
5 \( 1 + (-0.896 - 0.442i)T \)
7 \( 1 + (0.824 + 0.566i)T \)
11 \( 1 + (0.801 - 0.598i)T \)
13 \( 1 + (0.533 + 0.845i)T \)
17 \( 1 + (0.998 + 0.0523i)T \)
19 \( 1 + (-0.169 + 0.985i)T \)
23 \( 1 + (-0.996 + 0.0784i)T \)
29 \( 1 + (-0.908 - 0.418i)T \)
37 \( 1 + (-0.997 + 0.0654i)T \)
41 \( 1 + (-0.958 + 0.284i)T \)
43 \( 1 + (0.220 + 0.975i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (-0.999 + 0.0130i)T \)
59 \( 1 + (-0.639 - 0.768i)T \)
61 \( 1 + (-0.195 - 0.980i)T \)
67 \( 1 + (-0.751 + 0.659i)T \)
71 \( 1 + (-0.983 - 0.182i)T \)
73 \( 1 + (-0.333 + 0.942i)T \)
79 \( 1 + (-0.0523 + 0.998i)T \)
83 \( 1 + (-0.768 - 0.639i)T \)
89 \( 1 + (0.996 + 0.0784i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−17.97171420672040546233272576739, −17.59768982372333940721389898324, −16.87706998392693755256443338575, −16.19764380283645858012373594185, −15.425372874695499981565546326225, −14.87071502330952193953786502861, −14.06489065656656975797404042257, −13.349822461247780516387509516517, −12.33307615187283083366883485872, −11.997856039765578783123247562271, −11.29661599469885941588756808946, −10.6147145866892989676355822983, −10.26306972644095818216410669333, −9.06636812126598129077132395526, −8.11427591185464339450275180094, −7.47035638345964655127260278140, −7.07755760099989712586274497197, −6.1858561274516925295824515871, −5.377041898554567419771593697272, −4.57063370416635787403577556677, −3.97894456749992706565070049875, −3.152596700116936735880724829080, −1.828845569506115139552497822, −1.14311226351996890503128646042, −0.08571728224305342046276912995, 1.31988272926894097950971931489, 1.67324528986871904945398753753, 3.392896639567439293074429424212, 3.93280363732197306589749866958, 4.581098215655137139281716754723, 5.45694152811252566614154443447, 5.982228313910924034950365327339, 6.79743794087527657620204060005, 7.79386751717320959108398613531, 8.331262778053636065455737522714, 9.118986417475720482068162803710, 9.853639470047436731983630145239, 10.834999884859970135087469665308, 11.40828499182970971205849905310, 12.02665934238479942186852646146, 12.166521434788371647113215949444, 13.27203718491687406366505023008, 14.33050209516078074786603824574, 14.72017428740687618364382506096, 15.72381096864348654783377825616, 16.09830046734165584373935676669, 16.905921951889832364777532121873, 17.15079500951525512478429670509, 18.34865639723018959450375710153, 18.706797755258193456491890137705

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