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Properties

Label 1-317-317.175-r0-0-0
Degree $1$
Conductor $317$
Sign $0.862 - 0.506i$
Analytic cond. $1.47214$
Root an. cond. $1.47214$
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  + (−0.331 − 0.943i)2-s + (−0.610 + 0.792i)3-s + (−0.780 + 0.625i)4-s + (0.441 + 0.897i)5-s + (0.949 + 0.312i)6-s + (0.368 − 0.929i)7-s + (0.848 + 0.528i)8-s + (−0.255 − 0.966i)9-s + (0.700 − 0.714i)10-s + (−0.999 − 0.0397i)11-s + (−0.0198 − 0.999i)12-s + (−0.0992 − 0.995i)13-s + (−0.999 − 0.0397i)14-s + (−0.980 − 0.197i)15-s + (0.216 − 0.976i)16-s + (0.804 + 0.594i)17-s + ⋯
L(s)  = 1  + (−0.331 − 0.943i)2-s + (−0.610 + 0.792i)3-s + (−0.780 + 0.625i)4-s + (0.441 + 0.897i)5-s + (0.949 + 0.312i)6-s + (0.368 − 0.929i)7-s + (0.848 + 0.528i)8-s + (−0.255 − 0.966i)9-s + (0.700 − 0.714i)10-s + (−0.999 − 0.0397i)11-s + (−0.0198 − 0.999i)12-s + (−0.0992 − 0.995i)13-s + (−0.999 − 0.0397i)14-s + (−0.980 − 0.197i)15-s + (0.216 − 0.976i)16-s + (0.804 + 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.862 - 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8198432908 - 0.2228137652i\)
\(L(\frac12)\) \(\approx\) \(0.8198432908 - 0.2228137652i\)
\(L(1)\) \(\approx\) \(0.7674414891 - 0.1407735489i\)
\(L(1)\) \(\approx\) \(0.7674414891 - 0.1407735489i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.331 - 0.943i)T \)
3 \( 1 + (-0.610 + 0.792i)T \)
5 \( 1 + (0.441 + 0.897i)T \)
7 \( 1 + (0.368 - 0.929i)T \)
11 \( 1 + (-0.999 - 0.0397i)T \)
13 \( 1 + (-0.0992 - 0.995i)T \)
17 \( 1 + (0.804 + 0.594i)T \)
19 \( 1 + (0.949 - 0.312i)T \)
23 \( 1 + (-0.476 - 0.878i)T \)
29 \( 1 + (0.368 + 0.929i)T \)
31 \( 1 + (0.921 - 0.387i)T \)
37 \( 1 + (0.888 + 0.459i)T \)
41 \( 1 + (-0.869 + 0.494i)T \)
43 \( 1 + (0.987 - 0.158i)T \)
47 \( 1 + (0.921 - 0.387i)T \)
53 \( 1 + (0.949 - 0.312i)T \)
59 \( 1 + (0.987 + 0.158i)T \)
61 \( 1 + (-0.905 - 0.423i)T \)
67 \( 1 + (0.138 + 0.990i)T \)
71 \( 1 + (-0.980 + 0.197i)T \)
73 \( 1 + (0.971 - 0.236i)T \)
79 \( 1 + (-0.780 - 0.625i)T \)
83 \( 1 + (0.641 + 0.767i)T \)
89 \( 1 + (-0.331 - 0.943i)T \)
97 \( 1 + (0.888 + 0.459i)T \)
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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

−25.08616670294799252351047909911, −24.37986269394859027757953220749, −23.79137789376164031678588300534, −22.94655600382970521346896542573, −21.809400729866438378535111044001, −20.936879063466069063769387461468, −19.47983664126390709674731849440, −18.57163332341015453382696252473, −18.00500035744613498646153742323, −17.16680809556234265561689501568, −16.2520207437144410510436358138, −15.637392659025550964673126909702, −14.08745312467912472399389491273, −13.56001912992820132460357141434, −12.391804283409983986492809091380, −11.65763460248886434920326044069, −10.0978664716837529749903533360, −9.17166072827123491732138723273, −8.11194628116426223742708703851, −7.4320034157744249212116773494, −6.01078877174519127774427364885, −5.47242822220684597157239977880, −4.65924534268287528578626992926, −2.23609773972858078701571079634, −1.04585404852764625286391883313, 0.895192143223642479740455912594, 2.698821951908631722873437516554, 3.54805514719477704186195621315, 4.75378995201267979255756400081, 5.75008247108190792221266534139, 7.29259897837910378678286970251, 8.302425619068734137731532273135, 9.89354275261678686037333802481, 10.30518151648598530128912000950, 10.86206863431585100807289856525, 11.88932908549350546959060041249, 13.06467473062151969211328473538, 14.05321076372651100255705468491, 15.00895377011351544422409282008, 16.27690672351883822442329116208, 17.255790015785162705264771729978, 17.92616496914418962232255339996, 18.59872819687262922495517364845, 20.02264143184953289459511453341, 20.6812290198945007348307774441, 21.47807061113648898457472977609, 22.30217407777575891084007557023, 22.9747057742333900440280488429, 23.79008267135860429244091927421, 25.58602868487372492992754180818

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