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Properties

Label 1-805-805.562-r0-0-0
Degree $1$
Conductor $805$
Sign $0.977 - 0.213i$
Analytic cond. $3.73840$
Root an. cond. $3.73840$
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  + (0.371 + 0.928i)2-s + (−0.0950 + 0.995i)3-s + (−0.723 + 0.690i)4-s + (−0.959 + 0.281i)6-s + (−0.909 − 0.415i)8-s + (−0.981 − 0.189i)9-s + (−0.928 − 0.371i)11-s + (−0.618 − 0.786i)12-s + (−0.540 − 0.841i)13-s + (0.0475 − 0.998i)16-s + (−0.971 − 0.235i)17-s + (−0.189 − 0.981i)18-s + (0.235 + 0.971i)19-s − i·22-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.371 + 0.928i)2-s + (−0.0950 + 0.995i)3-s + (−0.723 + 0.690i)4-s + (−0.959 + 0.281i)6-s + (−0.909 − 0.415i)8-s + (−0.981 − 0.189i)9-s + (−0.928 − 0.371i)11-s + (−0.618 − 0.786i)12-s + (−0.540 − 0.841i)13-s + (0.0475 − 0.998i)16-s + (−0.971 − 0.235i)17-s + (−0.189 − 0.981i)18-s + (0.235 + 0.971i)19-s − i·22-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.977 - 0.213i$
Analytic conductor: \(3.73840\)
Root analytic conductor: \(3.73840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 805,\ (0:\ ),\ 0.977 - 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4483174885 - 0.04834628553i\)
\(L(\frac12)\) \(\approx\) \(0.4483174885 - 0.04834628553i\)
\(L(1)\) \(\approx\) \(0.6309654891 + 0.5042442326i\)
\(L(1)\) \(\approx\) \(0.6309654891 + 0.5042442326i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.371 + 0.928i)T \)
3 \( 1 + (-0.0950 + 0.995i)T \)
11 \( 1 + (-0.928 - 0.371i)T \)
13 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (-0.971 - 0.235i)T \)
19 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (0.189 - 0.981i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.458 - 0.888i)T \)
59 \( 1 + (-0.0475 - 0.998i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (-0.618 + 0.786i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.690 - 0.723i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (-0.755 + 0.654i)T \)
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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

−22.18846676967884758610959746892, −21.58551533031275972256744916238, −20.47805017700398177408827923822, −19.9003705109070700373023168474, −19.13518151636131205291388146937, −18.429852533125457229609779925735, −17.74482161023268308474924226935, −16.98935258001810183020566793504, −15.57339028789983498991124331192, −14.78100673242051629652574012287, −13.635063584258512060234463380403, −13.39413382288080946867609849572, −12.41509773238974759548289649564, −11.739370756701297383996506418488, −11.00800372516482129449606429994, −10.04352675724054646315595297336, −9.037784096622321881804700615591, −8.19060761405813365011939137887, −7.042806326682286369237439007439, −6.24684716180867868206137430247, −5.084031055109363643945444726818, −4.41003528233914317912912225805, −2.849394741910453778251353678813, −2.341812187540925362799322079868, −1.25070800996601928936212585081, 0.1890132093222911651524378551, 2.6416216950673679261969935636, 3.4549928656949643009236822737, 4.51179974484676703224701466744, 5.247427565189385615846708254030, 5.918139226276823973160045920011, 7.04588425390340672356646025735, 8.15361504799400702617428822422, 8.66641256299450300550225247683, 9.85004787101659850548967510046, 10.44606961713504651054300170931, 11.61550811632389588943166354669, 12.55258936592915950925843115971, 13.48819438508430006577549059038, 14.30655226632909161997598012633, 15.07604957645769379576923551466, 15.889553700752319036280932519644, 16.17836182074630906467340565017, 17.3713867172725554094500769474, 17.77010491698925116901141001064, 18.86903617204945795694731449587, 20.08088698349661431551243280056, 20.85310601670858170009787774090, 21.6320994555438327939366308024, 22.254079430831558213040703702072

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