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Properties

Label 1-51e2-2601.715-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.148 + 0.988i$
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  + (−0.153 + 0.988i)2-s + (−0.952 − 0.303i)4-s + (0.552 − 0.833i)5-s + (−0.696 − 0.717i)7-s + (0.445 − 0.895i)8-s + (0.739 + 0.673i)10-s + (0.213 + 0.976i)11-s + (−0.998 + 0.0615i)13-s + (0.816 − 0.577i)14-s + (0.816 + 0.577i)16-s + (0.932 + 0.361i)19-s + (−0.779 + 0.626i)20-s + (−0.998 + 0.0615i)22-s + (0.969 − 0.243i)23-s + (−0.389 − 0.920i)25-s + (0.0922 − 0.995i)26-s + ⋯
L(s)  = 1  + (−0.153 + 0.988i)2-s + (−0.952 − 0.303i)4-s + (0.552 − 0.833i)5-s + (−0.696 − 0.717i)7-s + (0.445 − 0.895i)8-s + (0.739 + 0.673i)10-s + (0.213 + 0.976i)11-s + (−0.998 + 0.0615i)13-s + (0.816 − 0.577i)14-s + (0.816 + 0.577i)16-s + (0.932 + 0.361i)19-s + (−0.779 + 0.626i)20-s + (−0.998 + 0.0615i)22-s + (0.969 − 0.243i)23-s + (−0.389 − 0.920i)25-s + (0.0922 − 0.995i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.148 + 0.988i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.148 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6806630698 + 0.7905998780i\)
\(L(\frac12)\) \(\approx\) \(0.6806630698 + 0.7905998780i\)
\(L(1)\) \(\approx\) \(0.8050479456 + 0.3044483202i\)
\(L(1)\) \(\approx\) \(0.8050479456 + 0.3044483202i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.153 + 0.988i)T \)
5 \( 1 + (0.552 - 0.833i)T \)
7 \( 1 + (-0.696 - 0.717i)T \)
11 \( 1 + (0.213 + 0.976i)T \)
13 \( 1 + (-0.998 + 0.0615i)T \)
19 \( 1 + (0.932 + 0.361i)T \)
23 \( 1 + (0.969 - 0.243i)T \)
29 \( 1 + (0.213 + 0.976i)T \)
31 \( 1 + (-0.998 + 0.0615i)T \)
37 \( 1 + (-0.982 + 0.183i)T \)
41 \( 1 + (-0.389 + 0.920i)T \)
43 \( 1 + (-0.908 - 0.417i)T \)
47 \( 1 + (0.969 + 0.243i)T \)
53 \( 1 + (-0.273 + 0.961i)T \)
59 \( 1 + (0.881 + 0.473i)T \)
61 \( 1 + (0.881 - 0.473i)T \)
67 \( 1 + (-0.153 - 0.988i)T \)
71 \( 1 + (-0.273 + 0.961i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (-0.779 + 0.626i)T \)
83 \( 1 + (-0.389 - 0.920i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.696 - 0.717i)T \)
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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

−19.07117457872570648313420734920, −18.78171594510912848573668925148, −17.88484276664293324308625108597, −17.30856725095690129861356870843, −16.52827576498875185440589401695, −15.55362296639705141465817355644, −14.722100916521149413431192934619, −14.04294511483020035846945148971, −13.35319339925075894380103562585, −12.74993934434164481811675476223, −11.73716500399398147531287816262, −11.41683838318604810498205344877, −10.41762741554614652607825868628, −9.85572682399708340743866197292, −9.20190531347893145464923568819, −8.57966870934486120513520365670, −7.42483101385273871766147083002, −6.71931340087505275883006708739, −5.562874620978044293492794735793, −5.25435775363950109813620845899, −3.81484049152783072976853115372, −3.1259509780245253879077810425, −2.59811438339774802155668538975, −1.76061920527390448162738719340, −0.43296930656364652367222493174, 0.895011988183681095222484567165, 1.753347991213551761035447208891, 3.150038418097177726656382613580, 4.14455079570053052345167771891, 4.9432605067151029325161237009, 5.37756086174517898139770182570, 6.48394721968688067954754786931, 7.1117603288412260296552968403, 7.63553378797596561680714865442, 8.76971931394145903669634020710, 9.32869215577955877500048461707, 9.95581351914720554211561094029, 10.47090814894203854813303023634, 11.96719128455411734517217765763, 12.70210259848670330809741356088, 13.12788871764027166409958859344, 14.03643515229688236775248905278, 14.52161948073642220198498364121, 15.43322592519893724912388458242, 16.15702901612930006773085414458, 16.912485974403366819249916991385, 17.08814443037629720310093529201, 17.94038678214079347665855953770, 18.656173075392378722696949925280, 19.67486557970976785125159314955

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