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Properties

Label 1-1259-1259.2-r1-0-0
Degree $1$
Conductor $1259$
Sign $0.871 - 0.491i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (0.999 + 0.00499i)2-s + (0.690 − 0.723i)3-s + (0.999 + 0.00998i)4-s + (0.928 + 0.370i)5-s + (0.693 − 0.720i)6-s + (0.335 + 0.942i)7-s + (0.999 + 0.0149i)8-s + (−0.0474 − 0.998i)9-s + (0.926 + 0.375i)10-s + (0.860 − 0.509i)11-s + (0.697 − 0.716i)12-s + (−0.968 − 0.247i)13-s + (0.330 + 0.943i)14-s + (0.909 − 0.416i)15-s + (0.999 + 0.0199i)16-s + (0.997 + 0.0698i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.00499i)2-s + (0.690 − 0.723i)3-s + (0.999 + 0.00998i)4-s + (0.928 + 0.370i)5-s + (0.693 − 0.720i)6-s + (0.335 + 0.942i)7-s + (0.999 + 0.0149i)8-s + (−0.0474 − 0.998i)9-s + (0.926 + 0.375i)10-s + (0.860 − 0.509i)11-s + (0.697 − 0.716i)12-s + (−0.968 − 0.247i)13-s + (0.330 + 0.943i)14-s + (0.909 − 0.416i)15-s + (0.999 + 0.0199i)16-s + (0.997 + 0.0698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.871 - 0.491i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ 0.871 - 0.491i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.410952653 - 2.208093260i\)
\(L(\frac12)\) \(\approx\) \(8.410952653 - 2.208093260i\)
\(L(1)\) \(\approx\) \(3.305006622 - 0.5330858319i\)
\(L(1)\) \(\approx\) \(3.305006622 - 0.5330858319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.999 + 0.00499i)T \)
3 \( 1 + (0.690 - 0.723i)T \)
5 \( 1 + (0.928 + 0.370i)T \)
7 \( 1 + (0.335 + 0.942i)T \)
11 \( 1 + (0.860 - 0.509i)T \)
13 \( 1 + (-0.968 - 0.247i)T \)
17 \( 1 + (0.997 + 0.0698i)T \)
19 \( 1 + (0.146 - 0.989i)T \)
23 \( 1 + (0.796 - 0.604i)T \)
29 \( 1 + (0.828 + 0.559i)T \)
31 \( 1 + (0.244 + 0.969i)T \)
37 \( 1 + (-0.997 + 0.0748i)T \)
41 \( 1 + (-0.524 - 0.851i)T \)
43 \( 1 + (-0.176 + 0.984i)T \)
47 \( 1 + (-0.946 + 0.323i)T \)
53 \( 1 + (-0.502 - 0.864i)T \)
59 \( 1 + (-0.999 + 0.0299i)T \)
61 \( 1 + (0.582 - 0.812i)T \)
67 \( 1 + (-0.884 + 0.465i)T \)
71 \( 1 + (0.982 + 0.188i)T \)
73 \( 1 + (0.112 + 0.993i)T \)
79 \( 1 + (0.936 + 0.351i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (0.553 - 0.832i)T \)
97 \( 1 + (-0.970 + 0.242i)T \)
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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

−20.9833036747057607385030350464, −20.47905658673519461512392517960, −19.74880549579934002248106032660, −19.04439908385764434270625117379, −17.47282248880160271101006360145, −16.77254344247676961402657905558, −16.5300995363532979742260882691, −15.153009632706971977441654865439, −14.69098837760363365032241064151, −13.86217390225065204579513150484, −13.66916649655797358858387457910, −12.483024632270804428044549114516, −11.77972669157589456230732913850, −10.6180500014519127831678712892, −9.96234263760669967189257381482, −9.45455545710511874605065284288, −8.11235174133377441512889725820, −7.37828398362923488711677360117, −6.4378506896911657681171163994, −5.30014756398463748862568350431, −4.73761163444695964238890385226, −3.94171535768812076449504696997, −3.09830828895228227722042048445, −1.98574718744408109424153055990, −1.29221989248270531577206973155, 1.14299813514401828857831082793, 1.93789533549859365510200746568, 2.906376252000687027557646719167, 3.212952524991160116036237791149, 4.850930708593238486235156302985, 5.474696297902837917524220449529, 6.565966160469213326227079290749, 6.851865795460866802894527441220, 8.04690841136896095893168599426, 8.91851962883953933752966292630, 9.76150557742131247257561042073, 10.82816655292053346298042402651, 11.84187752245558930078929512761, 12.39345342979278817247947591320, 13.10474886881227480589355365627, 14.06848060924311681700253472849, 14.44626543504584071003848868981, 14.962840014701191302732576813930, 15.9612115108437082980004410476, 17.14612295647793895915234515168, 17.68057820114908206266585004497, 18.76621136221035461239157523686, 19.32124052205562042892371656150, 20.08110638034444851734424742075, 21.12906988979873176658222176122

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