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Properties

Label 1-35e2-1225.53-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.804 + 0.593i$
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  + (−0.967 − 0.251i)2-s + (−0.379 − 0.925i)3-s + (0.873 + 0.486i)4-s + (0.134 + 0.990i)6-s + (−0.722 − 0.691i)8-s + (−0.712 + 0.701i)9-s + (0.712 + 0.701i)11-s + (0.119 − 0.992i)12-s + (−0.998 − 0.0448i)13-s + (0.525 + 0.850i)16-s + (−0.999 − 0.0149i)17-s + (0.866 − 0.5i)18-s + (−0.669 − 0.743i)19-s + (−0.512 − 0.858i)22-s + (0.800 − 0.599i)23-s + (−0.365 + 0.930i)24-s + ⋯
L(s)  = 1  + (−0.967 − 0.251i)2-s + (−0.379 − 0.925i)3-s + (0.873 + 0.486i)4-s + (0.134 + 0.990i)6-s + (−0.722 − 0.691i)8-s + (−0.712 + 0.701i)9-s + (0.712 + 0.701i)11-s + (0.119 − 0.992i)12-s + (−0.998 − 0.0448i)13-s + (0.525 + 0.850i)16-s + (−0.999 − 0.0149i)17-s + (0.866 − 0.5i)18-s + (−0.669 − 0.743i)19-s + (−0.512 − 0.858i)22-s + (0.800 − 0.599i)23-s + (−0.365 + 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.804 + 0.593i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.804 + 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07303871036 - 0.2220418731i\)
\(L(\frac12)\) \(\approx\) \(-0.07303871036 - 0.2220418731i\)
\(L(1)\) \(\approx\) \(0.5000394749 - 0.2091568651i\)
\(L(1)\) \(\approx\) \(0.5000394749 - 0.2091568651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.967 - 0.251i)T \)
3 \( 1 + (-0.379 - 0.925i)T \)
11 \( 1 + (0.712 + 0.701i)T \)
13 \( 1 + (-0.998 - 0.0448i)T \)
17 \( 1 + (-0.999 - 0.0149i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.800 - 0.599i)T \)
29 \( 1 + (0.995 - 0.0896i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.119 - 0.992i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.635 + 0.772i)T \)
53 \( 1 + (0.486 - 0.873i)T \)
59 \( 1 + (0.999 - 0.0299i)T \)
61 \( 1 + (0.193 + 0.981i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.538 - 0.842i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.351 + 0.936i)T \)
89 \( 1 + (-0.887 - 0.460i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−21.51818048180839098304378531363, −20.47994711523147001957685865241, −19.78628603327760937372372538313, −19.19200237089732577727009222744, −18.17985270838721762284762292580, −17.32739236658168322445346467287, −16.87116702365622054416772632901, −16.24550132602565538114811735879, −15.332044009125612211752659427456, −14.80036489217151723437239794559, −14.00827465129335102786338545324, −12.58407077562356783349296668517, −11.64138541713732571509594332877, −11.103348315976520254856840566329, −10.31298617639073239947653762347, −9.52943442664037327727190846207, −8.914955188944515231374383186885, −8.146622206503221276396453373929, −6.952714753597777386014612633412, −6.270921360908634485289675715829, −5.40187193590099702499234830918, −4.44612288836920973107876471334, −3.3459599260832685044296351547, −2.31122644445417919367006173206, −0.99360371551514175947119690938, 0.08914097323374996171352771229, 0.96285198437587731784616745799, 2.17363801361429285053410247641, 2.5392792176294821439680701571, 4.105177347269127765608732665709, 5.21055104141090038288686765162, 6.504339057294818507226530279782, 6.92758689396545289345041268550, 7.60568198756905333713292958925, 8.714061929418020192355292773253, 9.20731750794547228659864313653, 10.38009266888514289847819076132, 11.04569885376183853939492710493, 11.857617488534553373994665376, 12.540295742082822780451278771429, 13.108405063058879182641735180518, 14.39104473772540235034754289881, 15.11465220908334147747793784185, 16.17569537937932431692983458171, 17.00392389678465295748638608048, 17.58770999213333554688062875043, 17.97504305719414521427372394044, 19.05561180035143737047268005207, 19.621189240154052867950866362728, 20.01730648815187145876012863906

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