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Properties

Label 1-51e2-2601.284-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.725 + 0.688i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (0.288 − 0.957i)2-s + (−0.833 − 0.552i)4-s + (−0.975 + 0.221i)5-s + (−0.479 + 0.877i)7-s + (−0.769 + 0.638i)8-s + (−0.0692 + 0.997i)10-s + (−0.325 − 0.945i)11-s + (−0.417 − 0.908i)13-s + (0.701 + 0.712i)14-s + (0.389 + 0.920i)16-s + (0.228 − 0.973i)19-s + (0.935 + 0.354i)20-s + (−0.999 + 0.0384i)22-s + (−0.854 + 0.519i)23-s + (0.901 − 0.431i)25-s + (−0.990 + 0.138i)26-s + ⋯
L(s)  = 1  + (0.288 − 0.957i)2-s + (−0.833 − 0.552i)4-s + (−0.975 + 0.221i)5-s + (−0.479 + 0.877i)7-s + (−0.769 + 0.638i)8-s + (−0.0692 + 0.997i)10-s + (−0.325 − 0.945i)11-s + (−0.417 − 0.908i)13-s + (0.701 + 0.712i)14-s + (0.389 + 0.920i)16-s + (0.228 − 0.973i)19-s + (0.935 + 0.354i)20-s + (−0.999 + 0.0384i)22-s + (−0.854 + 0.519i)23-s + (0.901 − 0.431i)25-s + (−0.990 + 0.138i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.725 + 0.688i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.725 + 0.688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2092981000 + 0.08350880095i\)
\(L(\frac12)\) \(\approx\) \(0.2092981000 + 0.08350880095i\)
\(L(1)\) \(\approx\) \(0.5964130227 - 0.3470962461i\)
\(L(1)\) \(\approx\) \(0.5964130227 - 0.3470962461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.288 - 0.957i)T \)
5 \( 1 + (-0.975 + 0.221i)T \)
7 \( 1 + (-0.479 + 0.877i)T \)
11 \( 1 + (-0.325 - 0.945i)T \)
13 \( 1 + (-0.417 - 0.908i)T \)
19 \( 1 + (0.228 - 0.973i)T \)
23 \( 1 + (-0.854 + 0.519i)T \)
29 \( 1 + (-0.898 - 0.438i)T \)
31 \( 1 + (-0.0384 - 0.999i)T \)
37 \( 1 + (0.783 + 0.620i)T \)
41 \( 1 + (-0.667 - 0.744i)T \)
43 \( 1 + (-0.788 - 0.614i)T \)
47 \( 1 + (0.988 + 0.153i)T \)
53 \( 1 + (0.914 - 0.403i)T \)
59 \( 1 + (0.459 + 0.888i)T \)
61 \( 1 + (0.764 - 0.644i)T \)
67 \( 1 + (-0.881 - 0.473i)T \)
71 \( 1 + (-0.690 + 0.723i)T \)
73 \( 1 + (-0.862 + 0.506i)T \)
79 \( 1 + (-0.911 + 0.410i)T \)
83 \( 1 + (0.431 + 0.901i)T \)
89 \( 1 + (-0.995 + 0.0922i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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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

−19.192947036096279783735711139507, −18.49577249329659011438169296586, −17.76041122553125004151395953475, −16.83401950176054643847265307165, −16.32599423700909712605623620344, −15.99818261999178855104654243654, −14.8253011435105009057893995832, −14.62456149147147642714438253202, −13.64791394000727611415888709808, −12.893266164609251799044508628325, −12.27429291295678118631314318106, −11.66498042507983817370041989804, −10.43636461668761251724775257878, −9.79363216658092754674019102522, −8.9455230060605148129415384589, −8.09746394113082739182086817630, −7.353828531576140290317625322543, −7.050538438256061948236762938594, −6.11442712781486331598307861050, −5.0677346660167305056074071629, −4.297887592786813006717382135965, −3.91823851454031336912039777777, −2.961920143680559972722184160432, −1.52078777733006121632551217872, −0.09342254997698861960267836725, 0.753429760154543920064159042418, 2.255261322705838241763181936016, 2.87031925944371780133047842238, 3.534061634839498245523985294209, 4.31096438071338329243172343212, 5.45513222481362986884961320096, 5.7594301338860466576302856062, 6.99037741077465326363681476523, 8.02386424118840401382821787794, 8.57237069112141386820554345691, 9.42837771321839219867825473129, 10.18208119771044907652465144702, 10.97521983184955026859634358986, 11.72130774975205059856440199702, 12.00752907171783325654991556095, 13.104659804086050292931140934574, 13.363237049062754961611724504175, 14.49412487932498029301730966326, 15.33002209777992251241730548565, 15.52418738312690058567172900297, 16.57825068876908502539279436407, 17.55874210118536819566693872687, 18.502237616079462900048473795962, 18.742844467333289468750781866280, 19.55070471512152165799024339583

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