Properties

Label 1-51e2-2601.52-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.744 + 0.667i$
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  + (0.992 + 0.122i)2-s + (0.969 + 0.243i)4-s + (−0.153 + 0.988i)5-s + (0.816 − 0.577i)7-s + (0.932 + 0.361i)8-s + (−0.273 + 0.961i)10-s + (−0.696 + 0.717i)11-s + (−0.779 + 0.626i)13-s + (0.881 − 0.473i)14-s + (0.881 + 0.473i)16-s + (−0.602 − 0.798i)19-s + (−0.389 + 0.920i)20-s + (−0.779 + 0.626i)22-s + (−0.908 − 0.417i)23-s + (−0.952 − 0.303i)25-s + (−0.850 + 0.526i)26-s + ⋯
L(s)  = 1  + (0.992 + 0.122i)2-s + (0.969 + 0.243i)4-s + (−0.153 + 0.988i)5-s + (0.816 − 0.577i)7-s + (0.932 + 0.361i)8-s + (−0.273 + 0.961i)10-s + (−0.696 + 0.717i)11-s + (−0.779 + 0.626i)13-s + (0.881 − 0.473i)14-s + (0.881 + 0.473i)16-s + (−0.602 − 0.798i)19-s + (−0.389 + 0.920i)20-s + (−0.779 + 0.626i)22-s + (−0.908 − 0.417i)23-s + (−0.952 − 0.303i)25-s + (−0.850 + 0.526i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.744 + 0.667i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8169612930 + 2.135578035i\)
\(L(\frac12)\) \(\approx\) \(0.8169612930 + 2.135578035i\)
\(L(1)\) \(\approx\) \(1.564475893 + 0.6808411960i\)
\(L(1)\) \(\approx\) \(1.564475893 + 0.6808411960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.992 + 0.122i)T \)
5 \( 1 + (-0.153 + 0.988i)T \)
7 \( 1 + (0.816 - 0.577i)T \)
11 \( 1 + (-0.696 + 0.717i)T \)
13 \( 1 + (-0.779 + 0.626i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (-0.908 - 0.417i)T \)
29 \( 1 + (-0.696 + 0.717i)T \)
31 \( 1 + (-0.779 + 0.626i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (-0.952 + 0.303i)T \)
43 \( 1 + (-0.0307 + 0.999i)T \)
47 \( 1 + (-0.908 + 0.417i)T \)
53 \( 1 + (0.0922 + 0.995i)T \)
59 \( 1 + (0.650 - 0.759i)T \)
61 \( 1 + (0.650 + 0.759i)T \)
67 \( 1 + (0.992 - 0.122i)T \)
71 \( 1 + (0.0922 + 0.995i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (-0.389 + 0.920i)T \)
83 \( 1 + (-0.952 - 0.303i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (0.816 - 0.577i)T \)
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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

−19.25048426175916196024693878817, −18.53160649870487190392132716757, −17.53814837564211051057828700730, −16.79435235852351627898366120873, −16.13944039661145585241262027187, −15.42473076385558082795805366917, −14.82474717362514957454526728913, −14.11709424330652801907599927783, −13.162108006677922289242571558535, −12.81986724092707177709113111057, −11.891861934982762564403737972855, −11.56096429672765835427470128148, −10.571508709090200050862742575697, −9.84822970213613952984355904170, −8.756444262643959465947471984585, −7.92978600643975123983178033743, −7.59875131559980911386669933523, −6.17685185369222754048790503832, −5.407692289007059852460125628188, −5.21617849985587366282392216101, −4.152525326158639551887692090973, −3.49910934007074323232739186412, −2.21912877014330316280949337540, −1.8534421438162553164922539072, −0.427760726572332075540201487823, 1.65418559801746448763359789897, 2.311160203930895414366455525233, 3.099434230909867160421635580851, 4.13467264996164029259887067266, 4.65553490522593504015234895614, 5.41589820584162968359234333211, 6.54962684392815375861239414546, 7.05821383965922516801220148418, 7.63460322517558339631900068860, 8.41586501460981240149786013752, 9.87274740892603159780549748543, 10.4084822451656001913860618446, 11.24861588600078312445073737595, 11.6067867337757573513655537911, 12.63045553051665898708341129695, 13.237804480461602075516585666891, 14.2312599982529288470458600431, 14.52886477211667562180729431580, 15.12274774770273562030020538468, 15.88951544267587260550811117910, 16.73894136211754829662556788417, 17.47602900662538419938544658967, 18.16468373707360762240980882884, 18.97459241918838196972191933108, 20.065132720328880464595366899523

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