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Properties

Label 1-35e2-1225.102-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.475 - 0.879i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (0.0598 + 0.998i)2-s + (−0.850 − 0.525i)3-s + (−0.992 + 0.119i)4-s + (0.473 − 0.880i)6-s + (−0.178 − 0.983i)8-s + (0.447 + 0.894i)9-s + (−0.447 + 0.894i)11-s + (0.907 + 0.420i)12-s + (0.351 − 0.936i)13-s + (0.971 − 0.237i)16-s + (−0.800 − 0.599i)17-s + (−0.866 + 0.5i)18-s + (0.104 + 0.994i)19-s + (−0.919 − 0.393i)22-s + (−0.817 − 0.575i)23-s + (−0.365 + 0.930i)24-s + ⋯
L(s)  = 1  + (0.0598 + 0.998i)2-s + (−0.850 − 0.525i)3-s + (−0.992 + 0.119i)4-s + (0.473 − 0.880i)6-s + (−0.178 − 0.983i)8-s + (0.447 + 0.894i)9-s + (−0.447 + 0.894i)11-s + (0.907 + 0.420i)12-s + (0.351 − 0.936i)13-s + (0.971 − 0.237i)16-s + (−0.800 − 0.599i)17-s + (−0.866 + 0.5i)18-s + (0.104 + 0.994i)19-s + (−0.919 − 0.393i)22-s + (−0.817 − 0.575i)23-s + (−0.365 + 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.475 - 0.879i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.475 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3541850580 - 0.2112050704i\)
\(L(\frac12)\) \(\approx\) \(0.3541850580 - 0.2112050704i\)
\(L(1)\) \(\approx\) \(0.6107576877 + 0.2281469286i\)
\(L(1)\) \(\approx\) \(0.6107576877 + 0.2281469286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.0598 + 0.998i)T \)
3 \( 1 + (-0.850 - 0.525i)T \)
11 \( 1 + (-0.447 + 0.894i)T \)
13 \( 1 + (0.351 - 0.936i)T \)
17 \( 1 + (-0.800 - 0.599i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.817 - 0.575i)T \)
29 \( 1 + (-0.753 + 0.657i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.907 + 0.420i)T \)
41 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (0.967 + 0.251i)T \)
53 \( 1 + (-0.119 - 0.992i)T \)
59 \( 1 + (0.280 - 0.959i)T \)
61 \( 1 + (-0.873 + 0.486i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.635 + 0.772i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.266 + 0.963i)T \)
89 \( 1 + (0.163 - 0.986i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−21.28907890997933636846927539292, −20.44710413832737924032506890210, −19.57209296007364923471183201038, −18.78405582147337169208972819193, −18.09282131690521569473429188325, −17.342327322307204532148874570646, −16.624075213675324878255777003914, −15.70067496028857667217169661665, −14.973625056300836704213687875937, −13.7055571172648198582554138671, −13.34932036680358014135373949026, −12.24314826826095652230267581069, −11.486101298985507541050241499486, −11.04207835505332656429122957956, −10.311729376174258214301251037031, −9.34176255120498718829751888832, −8.8251525154671944792318347676, −7.63454155191379279559318498153, −6.22837970644197315451532072282, −5.73722428787179701354043506267, −4.53185378049755590062933001436, −4.10910251997931138471704984160, −3.03222178390723680019688256234, −1.91809906910838311285154684262, −0.7421744212591224978615770334, 0.13685037828055317341767344453, 1.240214527544182124152524297063, 2.57397780977637584675793162805, 4.03872670281313265086135915540, 4.845435942510604728829804245790, 5.63832454945928744102788426922, 6.31300478588737376557256985800, 7.2195524016016694367719293582, 7.80050246758828431193905592894, 8.63473689958894991995004620343, 9.88724397861673599211680742832, 10.42404346551527110010425952465, 11.55911261920373887000927697910, 12.590248305444697190997135316344, 12.91203461389964860811619533312, 13.85490409299652277537594498086, 14.68874028019141086714580439667, 15.731090117106149450358455155455, 16.06041145354429525764972576157, 17.06038917112160359284191468410, 17.735025532124112824774168030098, 18.22690252637017803021835838604, 18.83794362535108971836464205602, 20.02199142684309451930593528961, 20.85632911381322729519736632027

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