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Properties

Label 1-247-247.163-r1-0-0
Degree $1$
Conductor $247$
Sign $-0.959 + 0.281i$
Analytic cond. $26.5438$
Root an. cond. $26.5438$
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  + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.866 + 0.5i)11-s − 12-s + (−0.5 − 0.866i)14-s + i·15-s + (−0.5 + 0.866i)16-s − 17-s + i·18-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.866 + 0.5i)11-s − 12-s + (−0.5 − 0.866i)14-s + i·15-s + (−0.5 + 0.866i)16-s − 17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(26.5438\)
Root analytic conductor: \(26.5438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (1:\ ),\ -0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04804722242 + 0.3349412032i\)
\(L(\frac12)\) \(\approx\) \(0.04804722242 + 0.3349412032i\)
\(L(1)\) \(\approx\) \(0.5953500977 + 0.08971244723i\)
\(L(1)\) \(\approx\) \(0.5953500977 + 0.08971244723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + iT \)
71 \( 1 + iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 - iT \)
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

−25.44889734114917191736708376271, −24.44242683337702636102916011593, −23.97281280047063749164950595600, −22.99774398397077240853942897864, −21.79442952265714119033307598389, −20.639437684922198122967902637887, −19.59426118072822652462785553201, −18.45483060736305563248348688116, −17.999038911643161073110589375043, −17.31755896128892246748827087018, −16.40826302141708447359385791692, −15.13399151739287618722412989658, −13.93887657159431903027321222925, −13.45063186658147154769195431149, −11.743225145935863975207596865375, −10.813218797613155835833090891588, −10.21061395756848730209593387444, −8.64457774717753988067974273271, −7.77666857028964645148333173299, −6.82032810002835834961501929849, −5.93785160968364102466159516630, −4.98265209784142570172502560825, −2.4585852747021587784283432798, −1.5705749264580442584158121155, −0.146397630111241458145248003992, 1.572202437982562660292064811, 2.70049452023236106517519381384, 4.41944848414097927596540035846, 5.32136889351901780646212792560, 6.616601245857113025844447187907, 8.26348420495867266566137346491, 8.96630457511276143147512453707, 10.04637990422659469206567419014, 10.64726110309167873288414145104, 11.78767654701413116109782298773, 12.59009104485455260566525948300, 13.96417237131098395304745063808, 15.43377674295310710261753887902, 16.03763992248690189734042829564, 17.35965902658084434312775521531, 17.6500397307226880357528803931, 18.52601647424399472867853960921, 20.235588550269054950153088805834, 20.615311671726974189101566879856, 21.665609354420382357817231114144, 21.9512706065706258155068052, 23.56295214507322022768622706352, 24.667905648130562048176778095410, 25.595295416847227070407323350471, 26.45240483734343503318762806874

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