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Properties

Label 1-51e2-2601.205-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.358 - 0.933i$
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  + (0.881 + 0.473i)2-s + (0.552 + 0.833i)4-s + (0.816 + 0.577i)5-s + (−0.779 − 0.626i)7-s + (0.0922 + 0.995i)8-s + (0.445 + 0.895i)10-s + (−0.998 + 0.0615i)11-s + (−0.908 − 0.417i)13-s + (−0.389 − 0.920i)14-s + (−0.389 + 0.920i)16-s + (−0.850 − 0.526i)19-s + (−0.0307 + 0.999i)20-s + (−0.908 − 0.417i)22-s + (−0.153 + 0.988i)23-s + (0.332 + 0.943i)25-s + (−0.602 − 0.798i)26-s + ⋯
L(s)  = 1  + (0.881 + 0.473i)2-s + (0.552 + 0.833i)4-s + (0.816 + 0.577i)5-s + (−0.779 − 0.626i)7-s + (0.0922 + 0.995i)8-s + (0.445 + 0.895i)10-s + (−0.998 + 0.0615i)11-s + (−0.908 − 0.417i)13-s + (−0.389 − 0.920i)14-s + (−0.389 + 0.920i)16-s + (−0.850 − 0.526i)19-s + (−0.0307 + 0.999i)20-s + (−0.908 − 0.417i)22-s + (−0.153 + 0.988i)23-s + (0.332 + 0.943i)25-s + (−0.602 − 0.798i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08730593120 - 0.1270391359i\)
\(L(\frac12)\) \(\approx\) \(0.08730593120 - 0.1270391359i\)
\(L(1)\) \(\approx\) \(1.196325581 + 0.4454465030i\)
\(L(1)\) \(\approx\) \(1.196325581 + 0.4454465030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.881 + 0.473i)T \)
5 \( 1 + (0.816 + 0.577i)T \)
7 \( 1 + (-0.779 - 0.626i)T \)
11 \( 1 + (-0.998 + 0.0615i)T \)
13 \( 1 + (-0.908 - 0.417i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (-0.153 + 0.988i)T \)
29 \( 1 + (-0.998 + 0.0615i)T \)
31 \( 1 + (-0.908 - 0.417i)T \)
37 \( 1 + (-0.273 - 0.961i)T \)
41 \( 1 + (0.332 - 0.943i)T \)
43 \( 1 + (0.992 + 0.122i)T \)
47 \( 1 + (-0.153 - 0.988i)T \)
53 \( 1 + (0.932 - 0.361i)T \)
59 \( 1 + (-0.952 + 0.303i)T \)
61 \( 1 + (-0.952 - 0.303i)T \)
67 \( 1 + (0.881 - 0.473i)T \)
71 \( 1 + (0.932 - 0.361i)T \)
73 \( 1 + (-0.602 - 0.798i)T \)
79 \( 1 + (-0.0307 + 0.999i)T \)
83 \( 1 + (0.332 + 0.943i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (-0.779 - 0.626i)T \)
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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

−19.78618118529314187231192162156, −18.76034340365586296010390056802, −18.600039004109351237691245155370, −17.40260423303468695142501228975, −16.48037988860375064909488017841, −16.128010840375832661796437111679, −15.10664185664177580406780498089, −14.57560825549023834085241396819, −13.72989203184307956938493183385, −12.961113837112986942609873243094, −12.60425200968183953290266620113, −12.04528554748125080462810775898, −10.89463746187570016341660676852, −10.26645349342809548360285563365, −9.5767005854490767930762047910, −8.968373765216971225413159812408, −7.865967370947458311874426127489, −6.76626479878978258085779521855, −6.09908122990679498957247124643, −5.449784202503217479322033074227, −4.79190805943938884616132809226, −3.945477654068344062891695453369, −2.71547769355728276499786968544, −2.40270292391538898572559629716, −1.44864530816732538049489564886, 0.029481587471613216735216449048, 1.982426479805998794786366092589, 2.56593274716897450220771655845, 3.40197903006997355272471515877, 4.15745149414251168535855135938, 5.34630461575997762077883822096, 5.63931213651063123341118384883, 6.63436405263971954839145978933, 7.301565817247560361646601589243, 7.72654774615906559498683257856, 9.05312915199443343850610646099, 9.77726720353640935245909149161, 10.70912640840701544946900690568, 11.0297197385425474969219129478, 12.40870966071065354846030137155, 12.84282293935073282641373438892, 13.52637329337215171340727061569, 14.008804154413636293709812532104, 15.005848476809492734313774931895, 15.33301471171029133593950854815, 16.30030807554832563904148805971, 16.98052038703184456873188134965, 17.52867095037268525730296049073, 18.26334260653490915769929563100, 19.27299431379301694737182432233

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