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Properties

Label 1-35e2-1225.2-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: $\chi_{1225} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1971101035 + 0.3123636247i\)
\(L(\frac12)\) \(\approx\) \(-0.1971101035 + 0.3123636247i\)
\(L(1)\) \(\approx\) \(1.294531053 + 0.6808815557i\)
\(L(1)\) \(\approx\) \(1.294531053 + 0.6808815557i\)

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

−20.5239141306008006429667969700, −19.856127646911921100725218787460, −19.02053137334379120344616617029, −18.554653640450119105176231090948, −17.56659763337074441232554116164, −16.34899543016781010630035029045, −15.34844507726984140883962628881, −14.967791403788894828564707353334, −14.15138691062064797620278069436, −13.40322525679660594538496155813, −12.66316270445808778581016044916, −12.13226717227504762597808614017, −10.76679147781943478683397485677, −10.27427795885079751967344386507, −9.5755465822880252387237496683, −8.6562698635191301096742070681, −7.73988781501750424167605480451, −6.85979338345827884387114072450, −5.49316298806962690110872833518, −4.842719607010653733549548772325, −3.96143212435848254276088792693, −2.89177274452359677728729945045, −2.51231523122739834151248557088, −1.37560751705036357643057180672, −0.04461743928660629668353735536, 1.7135061935560763017652636096, 2.784937663361004112439156100715, 3.44771548414937364411539958988, 4.48646613120291089299469146310, 5.23415899544185962749580695696, 6.399259575498318461790235723167, 7.12030546513394360368034551577, 7.92285291996414966719781929858, 8.542963038781298542794161445832, 9.35983609811367255935759186472, 10.3399948982562217051013079219, 11.53213761194554754127682324166, 12.5528583997117563366261641194, 13.09642160034950214482035197480, 13.758073856476087443445202816673, 14.6621071645629919377739057863, 15.06940893180609329836035024375, 15.85603883040355886454161208376, 16.79709116196148964562801819706, 17.38611398369706284863271266036, 18.69679363384915903821875652985, 18.88078897019236379694971988105, 20.08733016966803490415715387403, 20.83621948870785451777907399029, 21.636828990913184477914073884412

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