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Properties

Label 1-2527-2527.395-r0-0-0
Degree $1$
Conductor $2527$
Sign $0.898 + 0.438i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (0.999 − 0.0183i)2-s + (−0.896 + 0.443i)3-s + (0.999 − 0.0367i)4-s + (0.435 − 0.900i)5-s + (−0.888 + 0.459i)6-s + (0.998 − 0.0550i)8-s + (0.606 − 0.794i)9-s + (0.418 − 0.908i)10-s + (−0.677 + 0.735i)11-s + (−0.879 + 0.475i)12-s + (0.690 − 0.723i)13-s + (0.00918 + 0.999i)15-s + (0.997 − 0.0734i)16-s + (0.467 + 0.883i)17-s + (0.592 − 0.805i)18-s + ⋯
L(s)  = 1  + (0.999 − 0.0183i)2-s + (−0.896 + 0.443i)3-s + (0.999 − 0.0367i)4-s + (0.435 − 0.900i)5-s + (−0.888 + 0.459i)6-s + (0.998 − 0.0550i)8-s + (0.606 − 0.794i)9-s + (0.418 − 0.908i)10-s + (−0.677 + 0.735i)11-s + (−0.879 + 0.475i)12-s + (0.690 − 0.723i)13-s + (0.00918 + 0.999i)15-s + (0.997 − 0.0734i)16-s + (0.467 + 0.883i)17-s + (0.592 − 0.805i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ 0.898 + 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.749342652 + 0.6351331872i\)
\(L(\frac12)\) \(\approx\) \(2.749342652 + 0.6351331872i\)
\(L(1)\) \(\approx\) \(1.715155689 + 0.1167896912i\)
\(L(1)\) \(\approx\) \(1.715155689 + 0.1167896912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.999 - 0.0183i)T \)
3 \( 1 + (-0.896 + 0.443i)T \)
5 \( 1 + (0.435 - 0.900i)T \)
11 \( 1 + (-0.677 + 0.735i)T \)
13 \( 1 + (0.690 - 0.723i)T \)
17 \( 1 + (0.467 + 0.883i)T \)
23 \( 1 + (0.0642 + 0.997i)T \)
29 \( 1 + (-0.209 + 0.977i)T \)
31 \( 1 + (-0.191 + 0.981i)T \)
37 \( 1 + (0.298 + 0.954i)T \)
41 \( 1 + (0.983 - 0.182i)T \)
43 \( 1 + (-0.995 - 0.0917i)T \)
47 \( 1 + (0.842 + 0.539i)T \)
53 \( 1 + (0.0459 - 0.998i)T \)
59 \( 1 + (-0.333 + 0.942i)T \)
61 \( 1 + (0.800 - 0.599i)T \)
67 \( 1 + (0.227 + 0.973i)T \)
71 \( 1 + (-0.919 + 0.393i)T \)
73 \( 1 + (-0.999 - 0.0367i)T \)
79 \( 1 + (0.995 + 0.0917i)T \)
83 \( 1 + (0.962 - 0.272i)T \)
89 \( 1 + (0.999 - 0.0367i)T \)
97 \( 1 + (0.957 - 0.289i)T \)
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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

−19.02416306052581965663412350849, −18.780914005287574438752582855770, −18.07846910556528962797016251606, −17.1092038904762148005095533069, −16.38278766617357561311354748887, −15.943965408159914145260781815128, −14.99203274094167410019336009761, −14.193118095405619584736635999869, −13.52510367427852861195431832274, −13.17005671025472126711047974909, −12.107776619228232095887302011715, −11.50440753129211539395733806437, −10.89013605783491951649094367834, −10.42020119141529150564018370815, −9.3867042879498295603850706818, −7.95732899552494654282379018255, −7.38108828850325135919215150472, −6.52896907462963922775881222883, −6.027499962374119845372440743121, −5.45804833164709730971617056844, −4.51265938259137194303690346880, −3.62809891342758370890023427541, −2.592517292348403865680889703940, −2.04989020720717013945354428377, −0.78441273810413073924351849346, 1.11499669121015632922628090318, 1.74517066213061656303199825581, 3.08674069652119334436092798687, 3.90120263412999635207805280557, 4.70986047236698738292877348841, 5.40360242968078837530354143096, 5.73635587896387187190394724787, 6.65090188882526768418937101217, 7.56225996932543111314124316584, 8.45925296292208992634824232311, 9.56913564357182121596001269248, 10.33357021803217684023836788842, 10.778931275046016145628604773, 11.77867164921058000711519413285, 12.40716827390932458134848925764, 12.97746098454507863173652959741, 13.44540143231014058616146549352, 14.643189807995433439825232272019, 15.22669818741704247199317096804, 16.07261481810000791004993571429, 16.29470292301096718108173549054, 17.42366564170698900242103405246, 17.63430107881334885853308332739, 18.735238348753634689405145179567, 19.8703192558246680298684377205

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