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Properties

Label 1-2624-2624.1163-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.995 - 0.0991i$
Analytic cond. $12.1858$
Root an. cond. $12.1858$
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  + (−0.382 + 0.923i)3-s + (−0.522 − 0.852i)5-s + (−0.309 − 0.951i)7-s + (−0.707 − 0.707i)9-s + (0.522 − 0.852i)11-s + (−0.649 + 0.760i)13-s + (0.987 − 0.156i)15-s + (−0.156 + 0.987i)17-s + (0.996 + 0.0784i)19-s + (0.996 + 0.0784i)21-s + (0.453 + 0.891i)23-s + (−0.453 + 0.891i)25-s + (0.923 − 0.382i)27-s + (0.852 − 0.522i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.522 − 0.852i)5-s + (−0.309 − 0.951i)7-s + (−0.707 − 0.707i)9-s + (0.522 − 0.852i)11-s + (−0.649 + 0.760i)13-s + (0.987 − 0.156i)15-s + (−0.156 + 0.987i)17-s + (0.996 + 0.0784i)19-s + (0.996 + 0.0784i)21-s + (0.453 + 0.891i)23-s + (−0.453 + 0.891i)25-s + (0.923 − 0.382i)27-s + (0.852 − 0.522i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.995 - 0.0991i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.995 - 0.0991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.069270247 - 0.05312641310i\)
\(L(\frac12)\) \(\approx\) \(1.069270247 - 0.05312641310i\)
\(L(1)\) \(\approx\) \(0.8249099419 + 0.02518735918i\)
\(L(1)\) \(\approx\) \(0.8249099419 + 0.02518735918i\)

Euler product

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

−19.42469291977060717733132695833, −18.33939054158470138877269826028, −18.24410822221356286354516742659, −17.50324876664962909463249087513, −16.57445245619839846020620347693, −15.680900089258113083017131754440, −15.17687062371428735363979388172, −14.291712965030600123501780487987, −13.78647165439868841913626657854, −12.54116360385576983239154098847, −12.26923531998541129403929590125, −11.69483538103413575116617330314, −10.8544442844720941887002468741, −10.03843882467313618798318967839, −9.18695692763781639458559399996, −8.24863079895782443943397537750, −7.51385295401031544962928094555, −6.81483769918730172975636620156, −6.40961430162970893472024509777, −5.25308362634286973185970891474, −4.756367372673040054017503364403, −3.19372207493268065025458038382, −2.78196528438738915924862802999, −1.92833820555330242704139076235, −0.65291998673649804215933732492, 0.610813043545362532999090941344, 1.47450550893131767353609533043, 3.13446901888267194646949681989, 3.73773249005918417527651867501, 4.41315374671978050065876303140, 5.04461780648550678092010621233, 5.99755355606325549617374580643, 6.75099554676127687268173613784, 7.75710309764238802927747678545, 8.526313059440024084426211222137, 9.33074656795060094955404015608, 9.85139415670631336769634671475, 10.704684511194858972917457663159, 11.54531123648677043111669046858, 11.8927288860666720186506149816, 12.860770934565548374700521408449, 13.8117468399267661436428721034, 14.24098216383615142705409046414, 15.45643105167878929607993021986, 15.750968755453177527997331270169, 16.666678118301800231773238537866, 17.058116253322544460705839660538, 17.41848002817665700559165063943, 18.87223228257655106272904443193, 19.56228270220102606120328109441

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