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Properties

Label 1-317-317.138-r0-0-0
Degree $1$
Conductor $317$
Sign $0.999 + 0.0125i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.999 + 0.0125i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.999 + 0.0125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9227967396 + 0.005768975782i\)
\(L(\frac12)\) \(\approx\) \(0.9227967396 + 0.005768975782i\)
\(L(1)\) \(\approx\) \(0.8119389354 + 0.1553821161i\)
\(L(1)\) \(\approx\) \(0.8119389354 + 0.1553821161i\)

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

−25.1453613493280897909096104634, −24.727360049728216240958233657095, −23.382701474830484231934157376326, −22.22582510457586414093462639823, −21.67626479731883586316205197012, −20.452233180214324526726992058396, −19.72766050622514948896508946273, −18.57598676030594513378789742776, −18.10685178911131881919046304706, −17.4071871073185718331138810969, −16.72108942153457392739146595490, −14.89315182477473700109277192083, −14.23722174236185862199869893239, −12.88403729043274635495443051782, −12.18855807508354897318742541019, −11.34481499769883875712669949084, −10.296316809023690450034481819507, −9.44500424965937194803179693167, −8.06745035796898261710452895339, −7.498739825988296239374666328292, −6.37166404709351500036073833645, −5.22271394789875741372803422437, −3.35133612893699510481237160309, −1.991604282027192066532122314104, −1.61733282469928448412261322360, 0.828876390468146550443419574968, 2.32300522045133792685273055661, 4.27779985520024694417028789404, 5.223161630707856019776163106625, 5.93857911117100206977360570650, 7.3392900947841891403401168694, 8.52883138180903349015735143602, 9.32992974919211324255434934, 9.98115664484352431953943210996, 11.10547277986332911430965710019, 11.83272261849031808054774163094, 13.84722375303890733979427956142, 14.28319486002666497297556960507, 15.33968340752995365346817404388, 16.57285673958524478781653500708, 16.81444029014532268339483108230, 17.640525792596504587904610961, 18.64463303558720050999654784801, 20.05475572950453319888598744722, 20.53048724109692454004681409050, 21.581657733542835010997440614882, 22.34814133339978600719502390082, 23.87738557502120703895390124929, 24.24254878171286578943774850971, 25.27610762681394515305800665384

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