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Properties

Label 1-3968-3968.1107-r0-0-0
Degree $1$
Conductor $3968$
Sign $-0.549 + 0.835i$
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  + (−0.169 − 0.985i)3-s + (−0.946 − 0.321i)5-s + (−0.942 − 0.333i)7-s + (−0.942 + 0.333i)9-s + (0.697 + 0.716i)11-s + (−0.995 − 0.0915i)13-s + (−0.156 + 0.987i)15-s + (0.838 + 0.544i)17-s + (0.296 − 0.955i)19-s + (−0.169 + 0.985i)21-s + (0.649 − 0.760i)23-s + (0.793 + 0.608i)25-s + (0.488 + 0.872i)27-s + (0.999 + 0.0392i)29-s + (0.587 − 0.809i)33-s + ⋯
L(s)  = 1  + (−0.169 − 0.985i)3-s + (−0.946 − 0.321i)5-s + (−0.942 − 0.333i)7-s + (−0.942 + 0.333i)9-s + (0.697 + 0.716i)11-s + (−0.995 − 0.0915i)13-s + (−0.156 + 0.987i)15-s + (0.838 + 0.544i)17-s + (0.296 − 0.955i)19-s + (−0.169 + 0.985i)21-s + (0.649 − 0.760i)23-s + (0.793 + 0.608i)25-s + (0.488 + 0.872i)27-s + (0.999 + 0.0392i)29-s + (0.587 − 0.809i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $-0.549 + 0.835i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ -0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1156385130 - 0.2146151252i\)
\(L(\frac12)\) \(\approx\) \(-0.1156385130 - 0.2146151252i\)
\(L(1)\) \(\approx\) \(0.5960629513 - 0.2930521911i\)
\(L(1)\) \(\approx\) \(0.5960629513 - 0.2930521911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.169 - 0.985i)T \)
5 \( 1 + (-0.946 - 0.321i)T \)
7 \( 1 + (-0.942 - 0.333i)T \)
11 \( 1 + (0.697 + 0.716i)T \)
13 \( 1 + (-0.995 - 0.0915i)T \)
17 \( 1 + (0.838 + 0.544i)T \)
19 \( 1 + (0.296 - 0.955i)T \)
23 \( 1 + (0.649 - 0.760i)T \)
29 \( 1 + (0.999 + 0.0392i)T \)
37 \( 1 + (-0.659 - 0.751i)T \)
41 \( 1 + (-0.999 + 0.0261i)T \)
43 \( 1 + (-0.768 - 0.639i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (-0.143 - 0.989i)T \)
59 \( 1 + (-0.220 + 0.975i)T \)
61 \( 1 + (-0.555 - 0.831i)T \)
67 \( 1 + (0.997 - 0.0654i)T \)
71 \( 1 + (-0.430 + 0.902i)T \)
73 \( 1 + (0.566 - 0.824i)T \)
79 \( 1 + (0.544 - 0.838i)T \)
83 \( 1 + (-0.975 + 0.220i)T \)
89 \( 1 + (-0.649 - 0.760i)T \)
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

−18.95321669301046544831444034812, −18.53457582539124226364523145980, −17.28684907096643431450817068605, −16.72395112895908759377964278951, −16.21070571611934708023898657015, −15.62052612401970870775490692106, −14.967782782631615724811724542949, −14.35770207883065617058147397030, −13.69588234202854095158104934023, −12.4683637253440751363068494939, −11.90966286483520996775218593921, −11.551441769209865803763595665260, −10.58625310303545153739897389256, −9.8828206591424458956677076269, −9.442281630185833250048253469373, −8.5565781577563599967309465809, −7.89031644726893286893907161239, −6.893120007898894569431145548157, −6.3073142975200094841007159323, −5.361627745114687527556525570748, −4.75805480500958473935215997573, −3.711502917242987658564101959855, −3.29449875853955171083434711152, −2.76014644786834211287042998625, −1.13309982978870886615636708767, 0.0957226808991819787575030553, 0.90129620884806510824479936048, 1.89428178018771813711361266667, 2.951631010190497784221887238016, 3.510469342957632299884890497580, 4.60516506120056799295202419634, 5.17285793405675485370131665743, 6.31066774682010213878387295241, 7.07591058016583947084505571997, 7.18690624101599473058398296427, 8.2258718281388670432160709573, 8.835948595073645454991133142833, 9.73611037273396715324363656458, 10.45801863626558357653352773701, 11.41741214531382913578960308095, 12.084004058607512483734524712695, 12.51042504131187522218765913506, 12.99666616057538180762650708180, 13.90633010950345767807298136633, 14.652243998212259938922011990868, 15.27108699188614675284817732362, 16.14119247547550770300078315064, 16.93240805760672824944998193058, 17.15006203909709267733027874763, 18.094796699957189606940893350639

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