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Properties

Label 1-2624-2624.995-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: $\chi_{2624} (995, \cdot )$
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

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

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