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Properties

Label 1-35e2-1225.17-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.993 - 0.111i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.701 + 0.712i)2-s + (0.611 + 0.791i)3-s + (−0.0149 + 0.999i)4-s + (−0.134 + 0.990i)6-s + (−0.722 + 0.691i)8-s + (−0.251 + 0.967i)9-s + (0.251 + 0.967i)11-s + (−0.800 + 0.599i)12-s + (0.998 − 0.0448i)13-s + (−0.999 − 0.0299i)16-s + (−0.486 + 0.873i)17-s + (−0.866 + 0.5i)18-s + (−0.978 − 0.207i)19-s + (−0.512 + 0.858i)22-s + (0.119 − 0.992i)23-s + (−0.988 − 0.149i)24-s + ⋯
L(s)  = 1  + (0.701 + 0.712i)2-s + (0.611 + 0.791i)3-s + (−0.0149 + 0.999i)4-s + (−0.134 + 0.990i)6-s + (−0.722 + 0.691i)8-s + (−0.251 + 0.967i)9-s + (0.251 + 0.967i)11-s + (−0.800 + 0.599i)12-s + (0.998 − 0.0448i)13-s + (−0.999 − 0.0299i)16-s + (−0.486 + 0.873i)17-s + (−0.866 + 0.5i)18-s + (−0.978 − 0.207i)19-s + (−0.512 + 0.858i)22-s + (0.119 − 0.992i)23-s + (−0.988 − 0.149i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.993 - 0.111i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1390890903 + 2.481447185i\)
\(L(\frac12)\) \(\approx\) \(-0.1390890903 + 2.481447185i\)
\(L(1)\) \(\approx\) \(1.044110155 + 1.383785982i\)
\(L(1)\) \(\approx\) \(1.044110155 + 1.383785982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.701 + 0.712i)T \)
3 \( 1 + (0.611 + 0.791i)T \)
11 \( 1 + (0.251 + 0.967i)T \)
13 \( 1 + (0.998 - 0.0448i)T \)
17 \( 1 + (-0.486 + 0.873i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.119 - 0.992i)T \)
29 \( 1 + (0.995 + 0.0896i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.800 - 0.599i)T \)
41 \( 1 + (-0.983 - 0.178i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.986 + 0.163i)T \)
53 \( 1 + (-0.999 - 0.0149i)T \)
59 \( 1 + (0.525 - 0.850i)T \)
61 \( 1 + (0.946 - 0.323i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (0.460 + 0.887i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.351 + 0.936i)T \)
89 \( 1 + (-0.842 - 0.538i)T \)
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

−20.65862208517079181392868705436, −20.04460504969143159505807606810, −19.26334761416648787224279275755, −18.66835621476252447896996092618, −18.10310612420332117335968883205, −16.97047675743632380839610413319, −15.813935042421482126721120851194, −15.13839100996800571418921699912, −14.2269116438632648291726279851, −13.51915843721616627146492718292, −13.26373330310336214365501354424, −12.176366106186848911702502834755, −11.48740125364191357361845383057, −10.83742237747243917460428652913, −9.6606264067975298273474418489, −8.87537663953783057410166756070, −8.17411079037323008781398913814, −6.88413569855403221006675683701, −6.260815742623496238565790447601, −5.431035184921968883212462295642, −4.12813643880623077392472467639, −3.4303410499985087846915024055, −2.58867602818761961135178153304, −1.6405901782462069614741692273, −0.695220275487527021304787296274, 1.90705728670944673012847544780, 2.8366140458345678473844678805, 3.92513266898286961679851139167, 4.34953073421840253300353209745, 5.24705331211695120579777765128, 6.33856111710482832118621731743, 6.99992102147676585996068449173, 8.354512952513194967491124474227, 8.48195039178700091001902579341, 9.57445780840373194565639120230, 10.60976091612454080503287796898, 11.29983759465615583892977906867, 12.636727565213286461118903268474, 12.98171430912409192464848494910, 14.12323830798964216834029736167, 14.58092913514727171576756393127, 15.38029163805105216230742565171, 15.85837098900195339581800875068, 16.74979476613570931310537819457, 17.41924742815326475921111874845, 18.31652531510323424183714865763, 19.4642518143556603936421278846, 20.29530581689584791306914058146, 20.82360558551229596225422264122, 21.72783022335739191328955665851

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