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Properties

Label 1-156-156.11-r1-0-0
Degree $1$
Conductor $156$
Sign $0.0386 - 0.999i$
Analytic cond. $16.7645$
Root an. cond. $16.7645$
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  − i·5-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)23-s − 25-s + (0.5 − 0.866i)29-s − i·31-s + (−0.5 − 0.866i)35-s + (0.866 + 0.5i)37-s + (0.866 + 0.5i)41-s + (−0.5 − 0.866i)43-s − i·47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  − i·5-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)23-s − 25-s + (0.5 − 0.866i)29-s − i·31-s + (−0.5 − 0.866i)35-s + (0.866 + 0.5i)37-s + (0.866 + 0.5i)41-s + (−0.5 − 0.866i)43-s − i·47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0386 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0386 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.0386 - 0.999i$
Analytic conductor: \(16.7645\)
Root analytic conductor: \(16.7645\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 156,\ (1:\ ),\ 0.0386 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.334778862 - 1.284208122i\)
\(L(\frac12)\) \(\approx\) \(1.334778862 - 1.284208122i\)
\(L(1)\) \(\approx\) \(1.138831060 - 0.3919890982i\)
\(L(1)\) \(\approx\) \(1.138831060 - 0.3919890982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 - iT \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−27.67113962149840126638225345257, −27.08979139809373743620505579826, −25.9673799704688274855355251197, −25.05479711541512101771622417510, −24.011105411815559266467009330367, −23.05137454393826763449880287869, −21.646847961874994288307354019665, −21.600497923858192418779729268339, −19.80589467029527176661782091364, −19.0615545462077578758306400401, −17.92455070081635889242063248568, −17.23504383467783964048856982464, −15.69850465752307845252500839262, −14.74740604000847928182574823630, −14.09373934404781253714895201221, −12.64860347591486832733774735793, −11.31211931664119275237675337305, −10.82369070313471641413168768282, −9.2525665118495382861034459938, −8.20707621423592877735568427024, −6.884566892584541230927668961780, −5.889230118608942467606540933834, −4.37055329964402383224340622511, −2.99761831503016915809995181432, −1.60694025295684615543315472113, 0.73121446386295199878008633438, 2.05205230701521941242040441323, 4.16198830126345009839868915573, 4.82381283766378956780658159967, 6.37563483511133006509457218633, 7.74332363329622503014486547552, 8.73904641997782183787548827690, 9.81369685686798854917612724944, 11.20976376481051509833823901404, 12.141490996120221043270877250969, 13.25658419933971127960794205709, 14.31413777349991439722871637494, 15.36464564228439699616928697441, 16.76589293726092598077008969851, 17.21203756716645336878322330065, 18.443972492305483844516697271143, 19.82943988487286962434311721628, 20.51072140967457522499389841714, 21.332330940025496116592092686728, 22.67676405456855800386879214139, 23.6175538617231003598873944669, 24.64151326929857082742427934397, 25.1751209342984891865674732430, 26.70250219483109893191568967212, 27.49733195392486343586431487977

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