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Properties

Label 1-3520-3520.1323-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.948 - 0.316i$
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  + (−0.852 − 0.522i)3-s + (0.987 − 0.156i)7-s + (0.453 + 0.891i)9-s + (0.0784 + 0.996i)13-s + (−0.309 − 0.951i)17-s + (0.852 + 0.522i)19-s + (−0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (0.0784 − 0.996i)27-s + (0.972 + 0.233i)29-s + (0.309 − 0.951i)31-s + (−0.522 − 0.852i)37-s + (0.453 − 0.891i)39-s + (0.156 − 0.987i)41-s + (−0.382 + 0.923i)43-s + ⋯
L(s)  = 1  + (−0.852 − 0.522i)3-s + (0.987 − 0.156i)7-s + (0.453 + 0.891i)9-s + (0.0784 + 0.996i)13-s + (−0.309 − 0.951i)17-s + (0.852 + 0.522i)19-s + (−0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (0.0784 − 0.996i)27-s + (0.972 + 0.233i)29-s + (0.309 − 0.951i)31-s + (−0.522 − 0.852i)37-s + (0.453 − 0.891i)39-s + (0.156 − 0.987i)41-s + (−0.382 + 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.948 - 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526796066 - 0.2480338946i\)
\(L(\frac12)\) \(\approx\) \(1.526796066 - 0.2480338946i\)
\(L(1)\) \(\approx\) \(0.9860715817 - 0.1187914061i\)
\(L(1)\) \(\approx\) \(0.9860715817 - 0.1187914061i\)

Euler product

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

−18.441877169293147991316369331089, −17.98409586693960225022060057852, −17.39804129539945627477338217898, −16.87759820838750286965953733889, −15.95804742010881445043600997998, −15.319885716063191622948888701946, −14.90576340300952674335245695048, −14.014167929830012773911530609732, −13.110828037736376183384036316124, −12.40403209368093898795884482711, −11.68831299802547707996443463036, −11.132941954933075429621449633970, −10.36266196581114890148793778965, −9.9953664251771975326704395527, −8.721173428451947247709582266402, −8.40790662424207499574594815392, −7.324369092371059209544726824182, −6.57047743598075259197678866606, −5.77175128972526900148738403465, −4.977221044299357104562551584637, −4.66047221779956144743346503685, −3.5738937794749735984077902876, −2.775228433464619961227934510390, −1.52588078936137061166915050425, −0.77017770357041003547353869541, 0.77354590306733137559267312215, 1.5654168191910060900725818074, 2.28590207979572914632050506300, 3.468568065660494632412999697563, 4.64487550056417026838054529025, 4.89369422114641619836276109780, 5.84914790421591190650889131721, 6.5787938705381543741813689068, 7.468748430026982310714651878348, 7.73255309159702537556770457753, 8.88656167568126336921455582093, 9.551304804395668663439402475211, 10.5828608976699383826440630749, 11.13007508642533442871819538247, 11.83553538900247696162013273180, 12.12089269170768979292011858968, 13.26245703360881789818183786864, 13.84941850655461991500436324477, 14.33039518109704589451571946598, 15.40901977162419261033083826077, 16.07359400222180757205844384559, 16.77966197222162060398882395383, 17.33783897219729906717764276694, 18.14699386666291726480569442210, 18.38895218727799595592652834690

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