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Properties

Label 1-35e2-1225.198-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.430 + 0.902i$
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  + (−0.611 − 0.791i)2-s + (−0.981 + 0.193i)3-s + (−0.251 + 0.967i)4-s + (0.753 + 0.657i)6-s + (0.919 − 0.393i)8-s + (0.925 − 0.379i)9-s + (−0.925 − 0.379i)11-s + (0.0598 − 0.998i)12-s + (0.722 − 0.691i)13-s + (−0.873 − 0.486i)16-s + (−0.701 − 0.712i)17-s + (−0.866 − 0.5i)18-s + (−0.913 − 0.406i)19-s + (0.266 + 0.963i)22-s + (−0.894 − 0.447i)23-s + (−0.826 + 0.563i)24-s + ⋯
L(s)  = 1  + (−0.611 − 0.791i)2-s + (−0.981 + 0.193i)3-s + (−0.251 + 0.967i)4-s + (0.753 + 0.657i)6-s + (0.919 − 0.393i)8-s + (0.925 − 0.379i)9-s + (−0.925 − 0.379i)11-s + (0.0598 − 0.998i)12-s + (0.722 − 0.691i)13-s + (−0.873 − 0.486i)16-s + (−0.701 − 0.712i)17-s + (−0.866 − 0.5i)18-s + (−0.913 − 0.406i)19-s + (0.266 + 0.963i)22-s + (−0.894 − 0.447i)23-s + (−0.826 + 0.563i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.430 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02685738324 + 0.01694775310i\)
\(L(\frac12)\) \(\approx\) \(0.02685738324 + 0.01694775310i\)
\(L(1)\) \(\approx\) \(0.4039843605 - 0.2043971345i\)
\(L(1)\) \(\approx\) \(0.4039843605 - 0.2043971345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.611 - 0.791i)T \)
3 \( 1 + (-0.981 + 0.193i)T \)
11 \( 1 + (-0.925 - 0.379i)T \)
13 \( 1 + (0.722 - 0.691i)T \)
17 \( 1 + (-0.701 - 0.712i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.894 - 0.447i)T \)
29 \( 1 + (0.0448 + 0.998i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.0598 - 0.998i)T \)
41 \( 1 + (-0.995 + 0.0896i)T \)
43 \( 1 + (0.974 - 0.222i)T \)
47 \( 1 + (-0.941 + 0.337i)T \)
53 \( 1 + (0.967 + 0.251i)T \)
59 \( 1 + (-0.0149 - 0.999i)T \)
61 \( 1 + (-0.772 - 0.635i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (0.959 - 0.280i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.178 + 0.983i)T \)
89 \( 1 + (-0.971 - 0.237i)T \)
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

−21.61475842525837894409998304307, −20.91763634495672354374473425639, −19.70388752550017537481387423337, −19.02571773325254472021410992582, −18.21142084523770237751563203160, −17.80351713027130384839842234159, −16.94479510284360643954128568173, −16.32894619308012811914241959821, −15.572337242959214492773081608341, −15.00885008852146719764603984224, −13.72046852266291927523575381581, −13.2075328674326615367772836905, −12.144319215396545901323948488563, −11.21764667595098645464862203373, −10.4497138683111796583955593792, −9.95160532131685918440778600544, −8.733196681120213800951483459751, −8.03155293446730233059605066875, −7.11454386287753876323300262654, −6.326869644967287549235694185153, −5.79429357697253217240211036365, −4.76122242687835358860355368452, −4.06370167464943235400845191995, −2.118384367395212785925912418742, −1.36513523424572777247471180372, 0.016547486425934313983358814400, 0.54115461355829669221956908030, 1.83147140437618389834892848018, 2.85936908678995678252675761992, 3.92914739253318090957317924911, 4.75907944126362324600218007938, 5.72711876818344307761696525207, 6.70273034272202641116891914732, 7.6694900819352471697784480206, 8.52652101246625377777214318620, 9.40050646538072803482261587238, 10.38974073742234390157843255132, 10.833163850544929941746229917004, 11.43137824037610332726298897527, 12.44581304358495378637642908549, 13.00226855866662721711071357698, 13.74210395803704876958044785600, 15.22915461369294161633424880713, 16.00659246567622617634515521074, 16.51503786698414649181416257927, 17.489722717948069735012193146016, 18.13783939164445507960870424246, 18.49127928839218908623382317365, 19.51399778628603422717592959509, 20.47435302803315755907156066308

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