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Properties

Label 1-51e2-2601.103-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.348 - 0.937i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (0.969 + 0.243i)2-s + (0.881 + 0.473i)4-s + (−0.952 − 0.303i)5-s + (0.332 − 0.943i)7-s + (0.739 + 0.673i)8-s + (−0.850 − 0.526i)10-s + (−0.0307 − 0.999i)11-s + (0.213 − 0.976i)13-s + (0.552 − 0.833i)14-s + (0.552 + 0.833i)16-s + (−0.273 + 0.961i)19-s + (−0.696 − 0.717i)20-s + (0.213 − 0.976i)22-s + (0.650 + 0.759i)23-s + (0.816 + 0.577i)25-s + (0.445 − 0.895i)26-s + ⋯
L(s)  = 1  + (0.969 + 0.243i)2-s + (0.881 + 0.473i)4-s + (−0.952 − 0.303i)5-s + (0.332 − 0.943i)7-s + (0.739 + 0.673i)8-s + (−0.850 − 0.526i)10-s + (−0.0307 − 0.999i)11-s + (0.213 − 0.976i)13-s + (0.552 − 0.833i)14-s + (0.552 + 0.833i)16-s + (−0.273 + 0.961i)19-s + (−0.696 − 0.717i)20-s + (0.213 − 0.976i)22-s + (0.650 + 0.759i)23-s + (0.816 + 0.577i)25-s + (0.445 − 0.895i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.134755836 - 1.484229374i\)
\(L(\frac12)\) \(\approx\) \(2.134755836 - 1.484229374i\)
\(L(1)\) \(\approx\) \(1.680253961 - 0.2431647320i\)
\(L(1)\) \(\approx\) \(1.680253961 - 0.2431647320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.969 + 0.243i)T \)
5 \( 1 + (-0.952 - 0.303i)T \)
7 \( 1 + (0.332 - 0.943i)T \)
11 \( 1 + (-0.0307 - 0.999i)T \)
13 \( 1 + (0.213 - 0.976i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.650 + 0.759i)T \)
29 \( 1 + (-0.0307 - 0.999i)T \)
31 \( 1 + (0.213 - 0.976i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (0.816 - 0.577i)T \)
43 \( 1 + (-0.998 - 0.0615i)T \)
47 \( 1 + (0.650 - 0.759i)T \)
53 \( 1 + (-0.982 + 0.183i)T \)
59 \( 1 + (-0.153 - 0.988i)T \)
61 \( 1 + (-0.153 + 0.988i)T \)
67 \( 1 + (0.969 - 0.243i)T \)
71 \( 1 + (-0.982 + 0.183i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (0.816 + 0.577i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (0.332 - 0.943i)T \)
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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

−19.58673313857161495898576850575, −18.95904956694155885767134928246, −18.32039944279992833936495418691, −17.39927517052712339567447927367, −16.25878224376848340952534717373, −15.81478483193173302852080300089, −15.07888850486629028414490817775, −14.61798752379457933313963278766, −13.982874702268279273841017485570, −12.68520415102044083268147124719, −12.5250782438339004430627047336, −11.64651813155106980674404781345, −11.13777062758996806307092805622, −10.459555197799439340648826768577, −9.29951897686794758703000276260, −8.63448516831018261948478667674, −7.556688091199677420380198602806, −6.86505878816586682148996382006, −6.335465410004017342783923086, −5.0049241767524845009337957778, −4.74363355114488558749378917725, −3.85331666868368643742094433327, −2.90508350740598921734597173163, −2.263948244705161948394956278664, −1.288746035006122609634127971071, 0.59276688712830268551234137848, 1.62543336057766259919141703931, 3.00883834582699376854778531511, 3.60686497657115361146829015084, 4.18598611071810621143996524205, 5.072186696069641234747130523456, 5.799313071304522542381285092693, 6.65516394314798214232912458968, 7.72084945275134475841591070026, 7.87140116721639205361165527086, 8.71508043702118646877194296694, 10.12431376907155189467093639213, 10.8475615212439299744652562188, 11.42819143912110760055843958834, 12.06350130390962172545412113529, 13.003505983372620670858939132527, 13.46062928794745853725026728746, 14.191731195211259228494440169052, 15.06436672987510024724689881462, 15.5289403899883062957054523657, 16.321499347596791311336264923251, 16.92629895828527736081189640981, 17.460899498927236281228669272450, 18.7868711101990291668305277462, 19.33971684472043863894695651665

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