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Properties

Label 1-247-247.151-r0-0-0
Degree $1$
Conductor $247$
Sign $-0.957 + 0.289i$
Analytic cond. $1.14706$
Root an. cond. $1.14706$
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  − i·2-s − 3-s − 4-s + i·5-s + i·6-s − i·7-s + i·8-s + 9-s + 10-s − i·11-s + 12-s − 14-s − i·15-s + 16-s − 17-s − i·18-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s + i·5-s + i·6-s − i·7-s + i·8-s + 9-s + 10-s − i·11-s + 12-s − 14-s − i·15-s + 16-s − 17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(1.14706\)
Root analytic conductor: \(1.14706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (0:\ ),\ -0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04469317497 - 0.3018407210i\)
\(L(\frac12)\) \(\approx\) \(-0.04469317497 - 0.3018407210i\)
\(L(1)\) \(\approx\) \(0.4621154256 - 0.3051148193i\)
\(L(1)\) \(\approx\) \(0.4621154256 - 0.3051148193i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - iT \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + iT \)
23 \( 1 + iT \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + T \)
43 \( 1 \)
47 \( 1 - T \)
53 \( 1 - iT \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - iT \)
71 \( 1 \)
73 \( 1 - iT \)
79 \( 1 + iT \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - iT \)
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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

−26.58757342977883269940346775426, −25.3375727507667486301360827776, −24.673539168811321105172583457391, −23.94632378634922585604759525573, −23.08639561310271289001902312256, −22.15388925607370920520513791399, −21.474999340952810506246492248327, −20.11421135010724137977378197633, −18.7600500896410430480380967131, −17.78405733153388845461161459388, −17.35336841526307239400386431984, −16.068481433440264649265343692437, −15.76977807281279742083432575281, −14.61550076453882295851848085802, −13.04073586981255851114154194665, −12.58190635099263603053359492102, −11.5503284855877715083810636798, −9.92281141209748104352072933504, −9.135953112003263724991816617822, −8.03937737543278444379154920068, −6.82026916311895559542312913875, −5.80504431044768559656103680487, −5.0052707563375591093389468709, −4.19017705301110812553975240557, −1.71078201359009922137744255212, 0.24547973942242378638566692489, 1.88653652492597693909088298138, 3.45535108605475969782896928437, 4.30984855013366960315626092301, 5.71524680413870982260637196421, 6.77661032419793351638537518184, 8.00951508293596102558315286844, 9.63712144498609403610376601835, 10.53991977440799962762414723146, 11.10733087097438360834718848550, 11.838497442697750310504998832781, 13.27860798056876563299970890139, 13.802101458110654715060118714615, 15.13671130465716163269837735429, 16.52232054392315741992697764702, 17.39158214627156837072815827501, 18.26335960241469549309057268228, 18.97227321183774058883182554253, 20.00361960754299928691685842672, 21.10062396566560554223361961223, 22.239692570294112777638958192642, 22.36162722052192206051678553863, 23.57353351597897855669638203479, 24.146440061218417637859210978577, 26.10227091665571910831359623583

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