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Properties

Label 1-317-317.150-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: $\chi_{317} (150, \cdot )$
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

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

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