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Properties

Label 1-3520-3520.1317-r0-0-0
Degree $1$
Conductor $3520$
Sign $-0.975 + 0.219i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
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  + (0.522 − 0.852i)3-s + (0.987 − 0.156i)7-s + (−0.453 − 0.891i)9-s + (−0.996 + 0.0784i)13-s + (−0.309 − 0.951i)17-s + (−0.522 + 0.852i)19-s + (0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (−0.996 − 0.0784i)27-s + (0.233 − 0.972i)29-s + (−0.309 + 0.951i)31-s + (0.852 − 0.522i)37-s + (−0.453 + 0.891i)39-s + (0.156 − 0.987i)41-s + (−0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (0.522 − 0.852i)3-s + (0.987 − 0.156i)7-s + (−0.453 − 0.891i)9-s + (−0.996 + 0.0784i)13-s + (−0.309 − 0.951i)17-s + (−0.522 + 0.852i)19-s + (0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (−0.996 − 0.0784i)27-s + (0.233 − 0.972i)29-s + (−0.309 + 0.951i)31-s + (0.852 − 0.522i)37-s + (−0.453 + 0.891i)39-s + (0.156 − 0.987i)41-s + (−0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.975 + 0.219i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ -0.975 + 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1011445537 - 0.9107960537i\)
\(L(\frac12)\) \(\approx\) \(-0.1011445537 - 0.9107960537i\)
\(L(1)\) \(\approx\) \(0.9677894375 - 0.4691286581i\)
\(L(1)\) \(\approx\) \(0.9677894375 - 0.4691286581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.522 - 0.852i)T \)
7 \( 1 + (0.987 - 0.156i)T \)
13 \( 1 + (-0.996 + 0.0784i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.522 + 0.852i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.233 - 0.972i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.852 - 0.522i)T \)
41 \( 1 + (0.156 - 0.987i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.649 + 0.760i)T \)
59 \( 1 + (-0.522 - 0.852i)T \)
61 \( 1 + (-0.760 + 0.649i)T \)
67 \( 1 + (0.923 - 0.382i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (0.156 + 0.987i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (-0.760 + 0.649i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−19.36026040964865614915739460127, −18.34702445486961830866720356194, −17.642601095004931830774665742138, −17.002565708068861168662735552362, −16.39806219599975526663727159971, −15.4200363904610703428256916969, −14.90056379221618500447892969081, −14.60821366424456361491963380157, −13.65962540590338591956528831078, −13.01870081407199887140992627976, −12.05630691665161999266821374425, −11.22373234550458068155426826938, −10.82918732858847718179049610959, −9.84001821447040737488103433159, −9.42600512100179203219452131042, −8.34120366170139235701655492572, −8.13668396897522967919913340680, −7.19476789129729806533012901407, −6.16464260154832201683300569542, −5.214637064978726733604280777698, −4.68697566208596176683587867900, −4.036659997586007495951641057910, −3.063106861409457696184709476134, −2.25541348989841786655714372305, −1.55057821233802518740264199485, 0.21948092094659962677904016907, 1.38353271960278718181111589623, 2.15231583045470999102805569080, 2.72106674857168040512197572609, 3.86261969157767151577221126712, 4.62018907618115373835738629378, 5.45487410350449546594579946201, 6.39852315653212177379388671166, 7.08131584639081633229416216745, 7.86706303698213874036863103125, 8.215992569236196960845024052969, 9.14050398198551489949406990462, 9.83667648444558928233869406114, 10.77277275781398058939110171022, 11.54606686728291114760185167805, 12.2652845752567584634103062766, 12.64902595944273172405893430044, 13.84680093352965240422880638667, 14.06470537250051518316660584660, 14.753118815415550351214596684309, 15.406978899296487314571964614328, 16.438589257688281325428433978054, 17.178245745718113004856082127796, 17.77701673637536406894269661551, 18.3945887815115899818934214669

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