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Properties

Label 1-35e2-1225.22-r1-0-0
Degree $1$
Conductor $1225$
Sign $0.916 + 0.400i$
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  + (−0.266 − 0.963i)2-s + (−0.990 − 0.134i)3-s + (−0.858 + 0.512i)4-s + (0.134 + 0.990i)6-s + (0.722 + 0.691i)8-s + (0.963 + 0.266i)9-s + (−0.963 + 0.266i)11-s + (0.919 − 0.393i)12-s + (0.998 + 0.0448i)13-s + (0.473 − 0.880i)16-s + (−0.512 + 0.858i)17-s − i·18-s + (−0.309 + 0.951i)19-s + (0.512 + 0.858i)22-s + (0.919 + 0.393i)23-s + (−0.623 − 0.781i)24-s + ⋯
L(s)  = 1  + (−0.266 − 0.963i)2-s + (−0.990 − 0.134i)3-s + (−0.858 + 0.512i)4-s + (0.134 + 0.990i)6-s + (0.722 + 0.691i)8-s + (0.963 + 0.266i)9-s + (−0.963 + 0.266i)11-s + (0.919 − 0.393i)12-s + (0.998 + 0.0448i)13-s + (0.473 − 0.880i)16-s + (−0.512 + 0.858i)17-s − i·18-s + (−0.309 + 0.951i)19-s + (0.512 + 0.858i)22-s + (0.919 + 0.393i)23-s + (−0.623 − 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9147216390 + 0.1909592969i\)
\(L(\frac12)\) \(\approx\) \(0.9147216390 + 0.1909592969i\)
\(L(1)\) \(\approx\) \(0.6286600748 - 0.1858527044i\)
\(L(1)\) \(\approx\) \(0.6286600748 - 0.1858527044i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.266 - 0.963i)T \)
3 \( 1 + (-0.990 - 0.134i)T \)
11 \( 1 + (-0.963 + 0.266i)T \)
13 \( 1 + (0.998 + 0.0448i)T \)
17 \( 1 + (-0.512 + 0.858i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.919 + 0.393i)T \)
29 \( 1 + (0.995 - 0.0896i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.919 - 0.393i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (-0.351 + 0.936i)T \)
53 \( 1 + (-0.512 - 0.858i)T \)
59 \( 1 + (-0.473 + 0.880i)T \)
61 \( 1 + (0.753 - 0.657i)T \)
67 \( 1 + (0.951 + 0.309i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.998 + 0.0448i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.351 - 0.936i)T \)
89 \( 1 + (0.0448 + 0.998i)T \)
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

−21.164762116902632655092300645950, −20.06436683681314549530178280938, −18.91334211721035330383402565076, −18.36455993333015377191552901477, −17.74255396942061605882349277345, −17.08467565520589791134160234727, −16.01911533247697799022206412969, −15.88934954990841636648882307493, −15.070685008154145906582797260799, −13.861162755775874861291509866, −13.24059929229061094721162353670, −12.4970747377978097335672971489, −11.16110107464086153486415365255, −10.779118188260106154034843375311, −9.82836267429969924218031159131, −8.91664080216536276342621984200, −8.14945458657957909159819357351, −7.0128058541721855323638290587, −6.57603533870170675942870590192, −5.580747854681146765593453687933, −4.94067956185827765145300281107, −4.19804391259103433352586019375, −2.7881724367715110011889533906, −1.100989852986202813538307839309, −0.375217740411623909872658644998, 0.795821364258129752014985353494, 1.65664322844032745071363919213, 2.69736041855510755378678589602, 3.93495339885126672570270593767, 4.59330394666277169546558642760, 5.62313414055452379368264453405, 6.39234630519498712436550589192, 7.67645178235828966931098017481, 8.26738581615830332490779632576, 9.42745436437460534434645161179, 10.22587677816667261340866126976, 10.95306331001415541152346829642, 11.34400077610067406894152502840, 12.4985038270951696824166660711, 12.90296517034078088128222074257, 13.58732461713802366590008025291, 14.80441648689927179526470134021, 15.871112407610836004810012342252, 16.512692425882710738003994182802, 17.573996232687676900160506188262, 17.81441935591810642941136032896, 18.82718031426399859319012139815, 19.17203622353191477383461701786, 20.36228333420008977769143141171, 21.17037954246836195733427646373

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