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Properties

Label 1-35e2-1225.36-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.430 - 0.902i$
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  + (0.134 + 0.990i)2-s + (0.753 − 0.657i)3-s + (−0.963 + 0.266i)4-s + (0.753 + 0.657i)6-s + (−0.393 − 0.919i)8-s + (0.134 − 0.990i)9-s + (0.134 + 0.990i)11-s + (−0.550 + 0.834i)12-s + (−0.691 − 0.722i)13-s + (0.858 − 0.512i)16-s + (−0.963 − 0.266i)17-s + 18-s + (−0.809 + 0.587i)19-s + (−0.963 + 0.266i)22-s + (−0.550 − 0.834i)23-s + (−0.900 − 0.433i)24-s + ⋯
L(s)  = 1  + (0.134 + 0.990i)2-s + (0.753 − 0.657i)3-s + (−0.963 + 0.266i)4-s + (0.753 + 0.657i)6-s + (−0.393 − 0.919i)8-s + (0.134 − 0.990i)9-s + (0.134 + 0.990i)11-s + (−0.550 + 0.834i)12-s + (−0.691 − 0.722i)13-s + (0.858 − 0.512i)16-s + (−0.963 − 0.266i)17-s + 18-s + (−0.809 + 0.587i)19-s + (−0.963 + 0.266i)22-s + (−0.550 − 0.834i)23-s + (−0.900 − 0.433i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.430 - 0.902i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2362109897 - 0.3743274425i\)
\(L(\frac12)\) \(\approx\) \(0.2362109897 - 0.3743274425i\)
\(L(1)\) \(\approx\) \(0.9256801112 + 0.1651513190i\)
\(L(1)\) \(\approx\) \(0.9256801112 + 0.1651513190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.134 + 0.990i)T \)
3 \( 1 + (0.753 - 0.657i)T \)
11 \( 1 + (0.134 + 0.990i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (-0.963 - 0.266i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.550 - 0.834i)T \)
29 \( 1 + (-0.0448 - 0.998i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.550 + 0.834i)T \)
41 \( 1 + (-0.995 + 0.0896i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.983 + 0.178i)T \)
53 \( 1 + (-0.963 + 0.266i)T \)
59 \( 1 + (0.858 - 0.512i)T \)
61 \( 1 + (0.936 - 0.351i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (-0.691 + 0.722i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (-0.691 + 0.722i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−21.43223310294933507478672727378, −20.64024468447057934819435733473, −19.69794645785472530278209317202, −19.48160654628931381134227855805, −18.66476624265860149687871514654, −17.64754283851330424740599155268, −16.79464371116670586149572821075, −15.93239439767015897594873705439, −14.92446319480333934340450262900, −14.36839343110767861327535952333, −13.537270330386720799142376384513, −13.02203909680296477583549439769, −11.86562212607859484465197000864, −11.06814515869434797508600878172, −10.52240039692721112520256073070, −9.49846050852866305140422904506, −8.95193815123592301797256575772, −8.32978534105916371533336711742, −7.14014884238055218746668669564, −5.826950906089299969876762293490, −4.86052847660725794427271550027, −4.09554302106654144169735339863, −3.38272017254457123466613132415, −2.41005123268128610239546097948, −1.683855301300841454530020201120, 0.13931279587583893808500239128, 1.76728131622684453960433184406, 2.729676539297407845518853489345, 3.90716564724252569530214780609, 4.63783569895271721337231635786, 5.75225012392416116892294352966, 6.74808580636374071324186653904, 7.186658232005169341355835462840, 8.14608714088955617491219917885, 8.670744046787858014299822744159, 9.65347487396855182910356714364, 10.32632062779515994223911792733, 11.98351957695772534998031797656, 12.561128678300992955779190742682, 13.234129921994999400665556060156, 14.037442666090499591475049218073, 14.85773160260753159183804710352, 15.19930115656980189829747172785, 16.12300277249878353522047921496, 17.4284834847562498164922832348, 17.495050519066066822108754873, 18.57341691952957958068030036440, 19.132892403764047415977571301129, 20.22989530823902777662870918029, 20.61713790169133231528605053647

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