Properties

Degree $1$
Conductor $6025$
Sign $0.120 + 0.992i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0523 + 0.998i)2-s + (−0.998 − 0.0523i)3-s + (−0.994 + 0.104i)4-s i·6-s + (−0.942 − 0.333i)7-s + (−0.156 − 0.987i)8-s + (0.994 + 0.104i)9-s + (0.333 − 0.942i)11-s + (0.998 − 0.0523i)12-s + (0.566 − 0.824i)13-s + (0.284 − 0.958i)14-s + (0.978 − 0.207i)16-s + (−0.760 − 0.649i)17-s + (−0.0523 + 0.998i)18-s + (0.958 − 0.284i)19-s + ⋯
L(s,χ)  = 1  + (0.0523 + 0.998i)2-s + (−0.998 − 0.0523i)3-s + (−0.994 + 0.104i)4-s i·6-s + (−0.942 − 0.333i)7-s + (−0.156 − 0.987i)8-s + (0.994 + 0.104i)9-s + (0.333 − 0.942i)11-s + (0.998 − 0.0523i)12-s + (0.566 − 0.824i)13-s + (0.284 − 0.958i)14-s + (0.978 − 0.207i)16-s + (−0.760 − 0.649i)17-s + (−0.0523 + 0.998i)18-s + (0.958 − 0.284i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.120 + 0.992i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.120 + 0.992i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.120 + 0.992i$
Motivic weight: \(0\)
Character: $\chi_{6025} (37, \cdot )$
Sato-Tate group: $\mu(240)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.120 + 0.992i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.4739779627 + 0.4199206673i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.4739779627 + 0.4199206673i\)
\(L(\chi,1)\) \(\approx\) \(0.5866620821 + 0.1996670584i\)
\(L(1,\chi)\) \(\approx\) \(0.5866620821 + 0.1996670584i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.812122099589734030052669827698, −16.87012838439536295550537345477, −16.45648906005066448515407557087, −15.70740486629315416737766509321, −14.892745629329441916934043470593, −14.18548431378967838348221804979, −13.19215849799440528065460879975, −12.86242845886085315927478342557, −12.153478186444215121806014423604, −11.73646038033142669052535583241, −10.98990042163188908515928291103, −10.29339610121921461111364172048, −9.83387996679127320097023238943, −9.07264747632900773918124469889, −8.57314001873513044064354263852, −7.21203664252720852176230391329, −6.72903107862094793985284181738, −5.902959212307432905764751397221, −5.26862781875841984987242674786, −4.40828328476600535593699170086, −3.89355368949270379384332225780, −3.14099501462131860202744060167, −1.92479279734985411218074410684, −1.60327895699061670863422942061, −0.32982881085554537094210887472, 0.59030127363762666679948388163, 1.285252944868658063739823671843, 2.94398838331814537213475605651, 3.63627754153682706464534595258, 4.2944938326401486831131276639, 5.29616664301104384363305077694, 5.727310376387143028113202766992, 6.3246620551446984249747179398, 7.00040711252036817530089638912, 7.52422448886818732537001757730, 8.414160376039641026986663252526, 9.22515908971616622362643802680, 9.846798299270336276001411471906, 10.42917136426736668042590439840, 11.44721749746341037004250408674, 11.79515063679138854496526472802, 12.95828537647662506327426524803, 13.356002112234148953133522113805, 13.65143102826514978768013773958, 14.75138727192809459254164149684, 15.66255846776890754389976511461, 15.92348327800337336686062571929, 16.42799636649595046143051117346, 17.149121728589480807874945396153, 17.69611295489979769914427204067

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