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Properties

Label 1-317-317.205-r0-0-0
Degree $1$
Conductor $317$
Sign $0.994 - 0.104i$
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  + (−0.476 − 0.878i)2-s + (0.949 + 0.312i)3-s + (−0.545 + 0.838i)4-s + (0.987 + 0.158i)5-s + (−0.177 − 0.984i)6-s + (−0.961 − 0.274i)7-s + (0.996 + 0.0794i)8-s + (0.804 + 0.594i)9-s + (−0.331 − 0.943i)10-s + (0.216 + 0.976i)11-s + (−0.780 + 0.625i)12-s + (0.971 − 0.236i)13-s + (0.216 + 0.976i)14-s + (0.888 + 0.459i)15-s + (−0.405 − 0.914i)16-s + (−0.936 + 0.350i)17-s + ⋯
L(s)  = 1  + (−0.476 − 0.878i)2-s + (0.949 + 0.312i)3-s + (−0.545 + 0.838i)4-s + (0.987 + 0.158i)5-s + (−0.177 − 0.984i)6-s + (−0.961 − 0.274i)7-s + (0.996 + 0.0794i)8-s + (0.804 + 0.594i)9-s + (−0.331 − 0.943i)10-s + (0.216 + 0.976i)11-s + (−0.780 + 0.625i)12-s + (0.971 − 0.236i)13-s + (0.216 + 0.976i)14-s + (0.888 + 0.459i)15-s + (−0.405 − 0.914i)16-s + (−0.936 + 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.994 - 0.104i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.994 - 0.104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454323065 - 0.07610463226i\)
\(L(\frac12)\) \(\approx\) \(1.454323065 - 0.07610463226i\)
\(L(1)\) \(\approx\) \(1.198689446 - 0.1641758412i\)
\(L(1)\) \(\approx\) \(1.198689446 - 0.1641758412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.476 - 0.878i)T \)
3 \( 1 + (0.949 + 0.312i)T \)
5 \( 1 + (0.987 + 0.158i)T \)
7 \( 1 + (-0.961 - 0.274i)T \)
11 \( 1 + (0.216 + 0.976i)T \)
13 \( 1 + (0.971 - 0.236i)T \)
17 \( 1 + (-0.936 + 0.350i)T \)
19 \( 1 + (-0.177 + 0.984i)T \)
23 \( 1 + (0.368 - 0.929i)T \)
29 \( 1 + (-0.961 + 0.274i)T \)
31 \( 1 + (0.578 - 0.815i)T \)
37 \( 1 + (-0.869 - 0.494i)T \)
41 \( 1 + (0.293 + 0.955i)T \)
43 \( 1 + (0.641 + 0.767i)T \)
47 \( 1 + (0.578 - 0.815i)T \)
53 \( 1 + (-0.177 + 0.984i)T \)
59 \( 1 + (0.641 - 0.767i)T \)
61 \( 1 + (-0.671 + 0.741i)T \)
67 \( 1 + (-0.0198 - 0.999i)T \)
71 \( 1 + (0.888 - 0.459i)T \)
73 \( 1 + (-0.255 - 0.966i)T \)
79 \( 1 + (-0.545 - 0.838i)T \)
83 \( 1 + (-0.0992 - 0.995i)T \)
89 \( 1 + (-0.476 - 0.878i)T \)
97 \( 1 + (-0.869 - 0.494i)T \)
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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

−25.373730665388047087147823487953, −24.49532086176641144089084079072, −23.89473015585878616625156314502, −22.55259074250887047408262926084, −21.67473570955495187636753357273, −20.60373527959662651023868729265, −19.44743907007216993909698188287, −18.92377985287019390239505941413, −17.99381891938452086118491068494, −17.11432869683456148624625169311, −15.910842930592351952903957338, −15.45469999155688305217952249386, −14.00245495888636902317805685803, −13.61217784660344816127707136484, −12.89599228058283093284848753362, −11.022043954047238845293531975873, −9.80228355590698877954146590456, −8.94252339532865222139016415214, −8.6874266231933869237364224950, −7.05344763787611965787537948463, −6.42542093499482288130433044471, −5.439935993311505565105735814818, −3.82462219428290173934970580965, −2.48705596821818317741610414136, −1.127348984801253621641949196263, 1.57824225644877690759389774485, 2.487920517086262790022744441286, 3.55418880893106402739369824509, 4.48297814180134308845211042603, 6.26247973963991273909029789422, 7.4341122902984527703418869352, 8.71166969109533108183317575117, 9.38780754105871780993267955278, 10.18362448586782863666967424738, 10.81363869949197940973283660855, 12.59431154635502130190789644243, 13.12431675390627146519927048932, 13.92484561972173885770120969341, 15.03521228102418308172146108799, 16.27217785355767839516542029401, 17.153357977995766433740158598187, 18.25729436890198097388020555511, 18.92674625307659390248780072186, 19.990421999354288724690033062687, 20.57459752974150901389927738311, 21.28115402848141043348453237720, 22.36475068989544111973016879352, 22.856072242327518044008868390887, 24.749680525849101330220362073080, 25.513988240615747923602164065789

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