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Properties

Label 1-317-317.179-r0-0-0
Degree $1$
Conductor $317$
Sign $-0.762 - 0.647i$
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  + (0.804 − 0.594i)2-s + (0.216 − 0.976i)3-s + (0.293 − 0.955i)4-s + (−0.780 + 0.625i)5-s + (−0.405 − 0.914i)6-s + (0.921 − 0.387i)7-s + (−0.331 − 0.943i)8-s + (−0.905 − 0.423i)9-s + (−0.255 + 0.966i)10-s + (0.511 − 0.859i)11-s + (−0.869 − 0.494i)12-s + (0.848 − 0.528i)13-s + (0.511 − 0.859i)14-s + (0.441 + 0.897i)15-s + (−0.827 − 0.561i)16-s + (−0.671 + 0.741i)17-s + ⋯
L(s)  = 1  + (0.804 − 0.594i)2-s + (0.216 − 0.976i)3-s + (0.293 − 0.955i)4-s + (−0.780 + 0.625i)5-s + (−0.405 − 0.914i)6-s + (0.921 − 0.387i)7-s + (−0.331 − 0.943i)8-s + (−0.905 − 0.423i)9-s + (−0.255 + 0.966i)10-s + (0.511 − 0.859i)11-s + (−0.869 − 0.494i)12-s + (0.848 − 0.528i)13-s + (0.511 − 0.859i)14-s + (0.441 + 0.897i)15-s + (−0.827 − 0.561i)16-s + (−0.671 + 0.741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $-0.762 - 0.647i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ -0.762 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6624267646 - 1.804490405i\)
\(L(\frac12)\) \(\approx\) \(0.6624267646 - 1.804490405i\)
\(L(1)\) \(\approx\) \(1.149062370 - 1.108351179i\)
\(L(1)\) \(\approx\) \(1.149062370 - 1.108351179i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.804 - 0.594i)T \)
3 \( 1 + (0.216 - 0.976i)T \)
5 \( 1 + (-0.780 + 0.625i)T \)
7 \( 1 + (0.921 - 0.387i)T \)
11 \( 1 + (0.511 - 0.859i)T \)
13 \( 1 + (0.848 - 0.528i)T \)
17 \( 1 + (-0.671 + 0.741i)T \)
19 \( 1 + (-0.405 + 0.914i)T \)
23 \( 1 + (-0.936 - 0.350i)T \)
29 \( 1 + (0.921 + 0.387i)T \)
31 \( 1 + (-0.610 - 0.792i)T \)
37 \( 1 + (0.987 + 0.158i)T \)
41 \( 1 + (0.641 + 0.767i)T \)
43 \( 1 + (-0.545 - 0.838i)T \)
47 \( 1 + (-0.610 - 0.792i)T \)
53 \( 1 + (-0.405 + 0.914i)T \)
59 \( 1 + (-0.545 + 0.838i)T \)
61 \( 1 + (0.368 + 0.929i)T \)
67 \( 1 + (0.888 + 0.459i)T \)
71 \( 1 + (0.441 - 0.897i)T \)
73 \( 1 + (0.996 - 0.0794i)T \)
79 \( 1 + (0.293 + 0.955i)T \)
83 \( 1 + (-0.727 + 0.685i)T \)
89 \( 1 + (0.804 - 0.594i)T \)
97 \( 1 + (0.987 + 0.158i)T \)
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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

−25.396043385892182407214510889903, −24.59712711422142501241770199821, −23.648773668337479706737391167157, −22.95362518820020833694810135043, −21.89932215608150698427492397923, −21.2097753868917537665489220244, −20.36961164803337632995027384263, −19.75400036297065579768282041901, −17.94582255286700185195279402085, −17.15724155335155268356405714586, −15.92602140512622706568522695029, −15.721191701523544606131375641795, −14.679605622537484226388493597251, −13.96978449961985046531619594954, −12.7099149773902025670228364588, −11.55526875610572228064309838758, −11.213731432108673085574741648484, −9.31884475124527590413572157182, −8.5810456188149614652011100540, −7.726012476226135256415393773970, −6.38995329999140550089765938787, −4.93104690976216831523256643021, −4.569869897708336344930152817901, −3.63466947591852874395251701800, −2.15230898564586234007902201747, 0.986109504847527795712368094639, 2.16030965810739230320759142192, 3.44154014723195949184651179154, 4.19207328316689376357689988895, 5.884472675412336483474211979461, 6.554963759252288577787707085288, 7.8584392314510363938547145433, 8.57570417753466529942813386338, 10.47339388781044746585142261275, 11.19660126724303943555482472203, 11.86201424475393018807692580184, 12.89266472771474150391371348030, 13.876844305387506980875703306634, 14.49512900684779330035028374767, 15.29657778216486446146141266972, 16.648098877335121904254461484499, 18.14185490445502547449676379944, 18.57782777125295349654021920527, 19.72595481459032705989797338983, 20.096465723202837099421956522477, 21.26102567651077594123239567237, 22.28281530642862909030997695257, 23.22020012044112191820146977358, 23.79314614512251550832417896924, 24.444625951023267291052333817301

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