Properties

Label 1-65e2-4225.413-r1-0-0
Degree $1$
Conductor $4225$
Sign $0.926 + 0.376i$
Analytic cond. $454.039$
Root an. cond. $454.039$
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  + (0.638 − 0.769i)2-s + (−0.990 − 0.136i)3-s + (−0.184 − 0.982i)4-s + (−0.737 + 0.675i)6-s + (0.600 + 0.799i)7-s + (−0.873 − 0.485i)8-s + (0.962 + 0.270i)9-s + (0.962 − 0.270i)11-s + (0.0483 + 0.998i)12-s + (0.998 + 0.0483i)14-s + (−0.932 + 0.362i)16-s + (0.0161 + 0.999i)17-s + (0.822 − 0.568i)18-s + (−0.104 + 0.994i)19-s + (−0.485 − 0.873i)21-s + (0.406 − 0.913i)22-s + ⋯
L(s)  = 1  + (0.638 − 0.769i)2-s + (−0.990 − 0.136i)3-s + (−0.184 − 0.982i)4-s + (−0.737 + 0.675i)6-s + (0.600 + 0.799i)7-s + (−0.873 − 0.485i)8-s + (0.962 + 0.270i)9-s + (0.962 − 0.270i)11-s + (0.0483 + 0.998i)12-s + (0.998 + 0.0483i)14-s + (−0.932 + 0.362i)16-s + (0.0161 + 0.999i)17-s + (0.822 − 0.568i)18-s + (−0.104 + 0.994i)19-s + (−0.485 − 0.873i)21-s + (0.406 − 0.913i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(454.039\)
Root analytic conductor: \(454.039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (1:\ ),\ 0.926 + 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.060750215 + 0.4029109103i\)
\(L(\frac12)\) \(\approx\) \(2.060750215 + 0.4029109103i\)
\(L(1)\) \(\approx\) \(1.109334263 - 0.3787721703i\)
\(L(1)\) \(\approx\) \(1.109334263 - 0.3787721703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.638 - 0.769i)T \)
3 \( 1 + (-0.990 - 0.136i)T \)
7 \( 1 + (0.600 + 0.799i)T \)
11 \( 1 + (0.962 - 0.270i)T \)
17 \( 1 + (0.0161 + 0.999i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.207 - 0.978i)T \)
29 \( 1 + (-0.369 + 0.929i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (0.999 - 0.00805i)T \)
41 \( 1 + (0.471 + 0.881i)T \)
43 \( 1 + (-0.316 + 0.948i)T \)
47 \( 1 + (-0.999 + 0.0241i)T \)
53 \( 1 + (0.732 - 0.681i)T \)
59 \( 1 + (0.541 + 0.840i)T \)
61 \( 1 + (-0.657 - 0.753i)T \)
67 \( 1 + (0.982 + 0.184i)T \)
71 \( 1 + (0.789 + 0.613i)T \)
73 \( 1 + (0.698 - 0.715i)T \)
79 \( 1 + (-0.0241 - 0.999i)T \)
83 \( 1 + (0.377 - 0.926i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.709 - 0.704i)T \)
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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

−17.778611180784051227078410478549, −17.3701839401038489784938262891, −16.82369455691260086606170999342, −16.22903812691250875792453956984, −15.41962446513781612770688696794, −14.961062817899726096130366388110, −13.94618807973378933915913365340, −13.63832154741188893920552629894, −12.733053591782022109757850365826, −12.021868983478556743288481955008, −11.35280505018015016667958641996, −11.01822080993129320106766428593, −9.77503779365440409739612942610, −9.27057955830078674503327245848, −8.25102279022763037831068679224, −7.21075436239029940265362130780, −7.0936037483051497209216235668, −6.27821994086446990838567601563, −5.32775067980507519715487264009, −4.909573891181410532656609696853, −4.11165320757556757423083947653, −3.62275041026846285089123549199, −2.336079235695205456423374106519, −1.17187965060603326284475448003, −0.342401872188225058932695384531, 0.85552358290616622248970052035, 1.56945231412813467314992509261, 2.1767728978040214681238563622, 3.34170976848885495156776412118, 4.173333304467999406732500743089, 4.73379735277428129590972445030, 5.60768422756153375597886680943, 6.124726793712733804169463909309, 6.62570381051272545122064659141, 7.86935706363610487970555571775, 8.664788714845917736334097609297, 9.5437493237530444032771151830, 10.18455433456968579988929272558, 11.113129131532943176458778767850, 11.34342367390740436025843196136, 12.09743388021852686124158660087, 12.695655692633295928262763639654, 13.11876961928503933298488050007, 14.34121010780757918021563400805, 14.70108737190936410541520317324, 15.30750020912866312516833252180, 16.43689843557793627904873266300, 16.76017829156394377196377373967, 17.83146249828280618557158299093, 18.305523935327533366236530626103

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