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Properties

Label 1-1259-1259.1042-r0-0-0
Degree $1$
Conductor $1259$
Sign $0.630 - 0.776i$
Analytic cond. $5.84677$
Root an. cond. $5.84677$
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  + (0.831 − 0.555i)2-s + (0.335 + 0.942i)3-s + (0.382 − 0.924i)4-s + (0.690 − 0.723i)5-s + (0.802 + 0.596i)6-s + (0.891 − 0.452i)7-s + (−0.196 − 0.980i)8-s + (−0.774 + 0.632i)9-s + (0.171 − 0.985i)10-s + (0.973 + 0.227i)11-s + (0.998 + 0.0499i)12-s + (−0.368 + 0.929i)13-s + (0.489 − 0.871i)14-s + (0.913 + 0.407i)15-s + (−0.707 − 0.706i)16-s + (−0.386 − 0.922i)17-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)2-s + (0.335 + 0.942i)3-s + (0.382 − 0.924i)4-s + (0.690 − 0.723i)5-s + (0.802 + 0.596i)6-s + (0.891 − 0.452i)7-s + (−0.196 − 0.980i)8-s + (−0.774 + 0.632i)9-s + (0.171 − 0.985i)10-s + (0.973 + 0.227i)11-s + (0.998 + 0.0499i)12-s + (−0.368 + 0.929i)13-s + (0.489 − 0.871i)14-s + (0.913 + 0.407i)15-s + (−0.707 − 0.706i)16-s + (−0.386 − 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ 0.630 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.272880270 - 1.558549604i\)
\(L(\frac12)\) \(\approx\) \(3.272880270 - 1.558549604i\)
\(L(1)\) \(\approx\) \(2.172550106 - 0.6008234606i\)
\(L(1)\) \(\approx\) \(2.172550106 - 0.6008234606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.831 - 0.555i)T \)
3 \( 1 + (0.335 + 0.942i)T \)
5 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (0.891 - 0.452i)T \)
11 \( 1 + (0.973 + 0.227i)T \)
13 \( 1 + (-0.368 + 0.929i)T \)
17 \( 1 + (-0.386 - 0.922i)T \)
19 \( 1 + (-0.107 - 0.994i)T \)
23 \( 1 + (0.344 + 0.938i)T \)
29 \( 1 + (0.524 - 0.851i)T \)
31 \( 1 + (0.622 + 0.782i)T \)
37 \( 1 + (-0.834 + 0.551i)T \)
41 \( 1 + (0.660 - 0.750i)T \)
43 \( 1 + (0.481 + 0.876i)T \)
47 \( 1 + (0.363 + 0.931i)T \)
53 \( 1 + (-0.793 + 0.608i)T \)
59 \( 1 + (-0.923 - 0.384i)T \)
61 \( 1 + (0.436 - 0.899i)T \)
67 \( 1 + (0.814 + 0.580i)T \)
71 \( 1 + (-0.919 + 0.393i)T \)
73 \( 1 + (-0.768 - 0.639i)T \)
79 \( 1 + (0.0224 + 0.999i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (-0.990 + 0.134i)T \)
97 \( 1 + (-0.822 - 0.568i)T \)
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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

−21.23640877898809100288348247903, −20.60642027632205865961987203558, −19.6705257790100862300153286463, −18.75834071029758040211714212145, −17.93806882626889994651730406204, −17.40225509510620793260252025492, −16.80952907813365066279764162597, −15.36357062776560972190196978540, −14.6512365832896314588774687936, −14.41211189863369556904660672124, −13.61995765152579382451254094151, −12.691380502511171357176993263519, −12.17252251452977538887160334451, −11.2626519690768836092412408313, −10.42440008531734591524021373471, −8.950936928374984755821194557524, −8.335244502872244350168658652771, −7.536033515776363405096067591798, −6.66273013836880778201342714002, −6.03547466430355875907040896698, −5.38591317532424601958068386397, −4.09969032663607248597281757457, −3.08948809112785457446218650991, −2.29705890248044275881553418793, −1.49422876163765289758295903213, 1.124464209841858585602364190155, 2.02777135093141173613530783253, 2.938233100819563929828435790013, 4.25051815412771578911667485586, 4.5799424076294778317667492672, 5.208075828886133060228784478047, 6.32135998245631609590433253261, 7.30548691538283910110303943466, 8.71608253182448443873663529927, 9.36781759454212889064888175486, 9.8874639045422722535456872940, 11.00395157556725869560087332641, 11.51313255872976617776093857925, 12.32383207028224279120535926004, 13.60805127199849247813912496687, 13.939436059171947426837439340647, 14.49208364014538699246547550950, 15.52033856321984345531758440268, 16.11320179641942824632260146075, 17.29179590124590460232351617500, 17.49788707983475005455055081959, 19.1121511070206861461872103329, 19.76297343143668436278011442814, 20.44709188583056551838587452964, 21.01156911473097702276827712143

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