Properties

Label 1-967-967.72-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.925 + 0.378i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.854 − 0.519i)2-s + (−0.917 − 0.398i)3-s + (0.460 − 0.887i)4-s + (−0.576 − 0.816i)5-s + (−0.990 + 0.136i)6-s + (−0.334 − 0.942i)7-s + (−0.0682 − 0.997i)8-s + (0.682 + 0.730i)9-s + (−0.917 − 0.398i)10-s + (0.682 + 0.730i)11-s + (−0.775 + 0.631i)12-s + (0.682 − 0.730i)13-s + (−0.775 − 0.631i)14-s + (0.203 + 0.979i)15-s + (−0.576 − 0.816i)16-s + (0.203 − 0.979i)17-s + ⋯
L(s)  = 1  + (0.854 − 0.519i)2-s + (−0.917 − 0.398i)3-s + (0.460 − 0.887i)4-s + (−0.576 − 0.816i)5-s + (−0.990 + 0.136i)6-s + (−0.334 − 0.942i)7-s + (−0.0682 − 0.997i)8-s + (0.682 + 0.730i)9-s + (−0.917 − 0.398i)10-s + (0.682 + 0.730i)11-s + (−0.775 + 0.631i)12-s + (0.682 − 0.730i)13-s + (−0.775 − 0.631i)14-s + (0.203 + 0.979i)15-s + (−0.576 − 0.816i)16-s + (0.203 − 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2811680374 - 1.431505131i\)
\(L(\frac12)\) \(\approx\) \(-0.2811680374 - 1.431505131i\)
\(L(1)\) \(\approx\) \(0.7269479497 - 0.9232911014i\)
\(L(1)\) \(\approx\) \(0.7269479497 - 0.9232911014i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.854 - 0.519i)T \)
3 \( 1 + (-0.917 - 0.398i)T \)
5 \( 1 + (-0.576 - 0.816i)T \)
7 \( 1 + (-0.334 - 0.942i)T \)
11 \( 1 + (0.682 + 0.730i)T \)
13 \( 1 + (0.682 - 0.730i)T \)
17 \( 1 + (0.203 - 0.979i)T \)
19 \( 1 + (-0.917 - 0.398i)T \)
23 \( 1 + (0.460 - 0.887i)T \)
29 \( 1 + (0.682 - 0.730i)T \)
31 \( 1 + (-0.0682 - 0.997i)T \)
37 \( 1 + (0.682 - 0.730i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (0.203 + 0.979i)T \)
47 \( 1 + (0.203 + 0.979i)T \)
53 \( 1 + (0.460 - 0.887i)T \)
59 \( 1 + (0.460 + 0.887i)T \)
61 \( 1 + (-0.990 + 0.136i)T \)
67 \( 1 + (-0.0682 + 0.997i)T \)
71 \( 1 + (0.854 + 0.519i)T \)
73 \( 1 + (0.460 + 0.887i)T \)
79 \( 1 + (0.460 + 0.887i)T \)
83 \( 1 + (-0.990 + 0.136i)T \)
89 \( 1 + (-0.0682 + 0.997i)T \)
97 \( 1 + T \)
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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

−22.09143328718157577322097153768, −21.65754008721268597826253859271, −21.27837652261642099207188523279, −19.812427567505629793738754002505, −18.91071295361757719847502031204, −18.25707057209798144749525079570, −17.12565638942947117174852810012, −16.55340381769452747416926126419, −15.7006422155498326358228016151, −15.211100087799764225282636289, −14.47317548222625582819633705232, −13.489506097523370743659157037061, −12.33271121363009942216417638926, −11.93415242549728113000908955727, −11.12659329304972175500170580662, −10.474477173196970821021137008716, −9.013902173283614287756727018587, −8.30568507649237562424604032985, −6.93767544013389212624234360602, −6.37185207276146559613677568955, −5.84981242540841075102585089868, −4.76580994117666095651839451981, −3.689067317489778585713162040, −3.32679470385514030261289991861, −1.76573601730204426100746924395, 0.59543364519907416362159507627, 1.22280610237723474859725330669, 2.60998855137458535558016826380, 4.07467375339342957608527078443, 4.396236115678872492747608086151, 5.34958640947587355080162489977, 6.37984880611446323695788414478, 7.03306095909432686312309520145, 7.99696025695819050726631098808, 9.41953418332100765775791553020, 10.26287761045264802389452686763, 11.13746708046780401836634187550, 11.70046105321865227562509752469, 12.68776289576715881830438425099, 12.97390744338202747841704492223, 13.81895005505401255162659791692, 14.966637263307373216453827521659, 15.825492817419906951095876107133, 16.53182150527537831318053274571, 17.2190337721114949815563126092, 18.2331588691162314488705694729, 19.23140451487692525400734745379, 19.84905823781670674349522560273, 20.55577940564741939652437604924, 21.21678430329164449615536592442

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