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Properties

Label 1-65e2-4225.1123-r0-0-0
Degree $1$
Conductor $4225$
Sign $-0.862 - 0.506i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (0.861 − 0.506i)2-s + (0.548 + 0.836i)3-s + (0.485 − 0.873i)4-s + (0.896 + 0.443i)6-s + (0.568 − 0.822i)7-s + (−0.0241 − 0.999i)8-s + (−0.399 + 0.916i)9-s + (−0.916 + 0.399i)11-s + (0.997 − 0.0724i)12-s + (0.0724 − 0.997i)14-s + (−0.527 − 0.849i)16-s + (−0.999 + 0.0241i)17-s + (0.120 + 0.992i)18-s + (−0.587 − 0.809i)19-s + (0.999 + 0.0241i)21-s + (−0.587 + 0.809i)22-s + ⋯
L(s)  = 1  + (0.861 − 0.506i)2-s + (0.548 + 0.836i)3-s + (0.485 − 0.873i)4-s + (0.896 + 0.443i)6-s + (0.568 − 0.822i)7-s + (−0.0241 − 0.999i)8-s + (−0.399 + 0.916i)9-s + (−0.916 + 0.399i)11-s + (0.997 − 0.0724i)12-s + (0.0724 − 0.997i)14-s + (−0.527 − 0.849i)16-s + (−0.999 + 0.0241i)17-s + (0.120 + 0.992i)18-s + (−0.587 − 0.809i)19-s + (0.999 + 0.0241i)21-s + (−0.587 + 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-0.862 - 0.506i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ -0.862 - 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4539137396 - 1.669589698i\)
\(L(\frac12)\) \(\approx\) \(0.4539137396 - 1.669589698i\)
\(L(1)\) \(\approx\) \(1.557472150 - 0.4815797601i\)
\(L(1)\) \(\approx\) \(1.557472150 - 0.4815797601i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.861 - 0.506i)T \)
3 \( 1 + (0.548 + 0.836i)T \)
7 \( 1 + (0.568 - 0.822i)T \)
11 \( 1 + (-0.916 + 0.399i)T \)
17 \( 1 + (-0.999 + 0.0241i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (-0.215 - 0.976i)T \)
31 \( 1 + (0.896 + 0.443i)T \)
37 \( 1 + (0.715 + 0.698i)T \)
41 \( 1 + (-0.0483 + 0.998i)T \)
43 \( 1 + (0.464 - 0.885i)T \)
47 \( 1 + (-0.681 + 0.732i)T \)
53 \( 1 + (-0.331 - 0.943i)T \)
59 \( 1 + (-0.997 + 0.0724i)T \)
61 \( 1 + (0.958 - 0.285i)T \)
67 \( 1 + (-0.485 - 0.873i)T \)
71 \( 1 + (-0.548 - 0.836i)T \)
73 \( 1 + (-0.399 - 0.916i)T \)
79 \( 1 + (0.681 - 0.732i)T \)
83 \( 1 + (0.836 + 0.548i)T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (-0.926 - 0.377i)T \)
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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

−18.59321854409664494257526943484, −17.86561031514790382003765271931, −17.525610101969785166581630279664, −16.40826541861741453983310634839, −15.77640486876366063080544748663, −15.112509010855940917679326618252, −14.567338350013737443085644551221, −13.90784712442123749325807132905, −13.2847300971509524994985800575, −12.65276740350630938166780465061, −12.12352980821145219713659782604, −11.38382830102052690724355208690, −10.71562645113004542987147252686, −9.449890727378581328815144080522, −8.52712363654258238493370936222, −8.21980869214595393349442479343, −7.55263801494262365772053621656, −6.7397057785718668572427768513, −5.916273248885560581661884844077, −5.56291038217436726248817593222, −4.502677093951567553984205816844, −3.7638525827451817521848850002, −2.70994072363617746902746928051, −2.355968349731443951677320243214, −1.50278473317529759201188335057, 0.265610661088587973632303757123, 1.71360872879063886734083503094, 2.38199898069558370135000560917, 3.05820088645023316606998653284, 4.036037286231624703829267140501, 4.63758768458694160464347542876, 4.85441475337817398045155521753, 6.01606856468876903995504790955, 6.78709638155212905032295656890, 7.778610370355968534653260855675, 8.3153231917360181032305020573, 9.41947408630388692385653209532, 10.00490164550086416699172956196, 10.68503674272436413509105381764, 11.0737483216588180932803985465, 11.85367506587519225388278734801, 12.8806450271616601232673590574, 13.56262716775513781327797970506, 13.80197846648212830974252359985, 14.8283763118395280739937382007, 15.15263315360123008031917604624, 15.846213066343224478177323457161, 16.47411774791172351271852321157, 17.44835543313221847652452084675, 18.07387916932347450784612586041

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