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Properties

Label 1-847-847.716-r0-0-0
Degree $1$
Conductor $847$
Sign $0.328 + 0.944i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (0.580 − 0.814i)2-s + (−0.5 + 0.866i)3-s + (−0.327 − 0.945i)4-s + (0.723 + 0.690i)5-s + (0.415 + 0.909i)6-s + (−0.959 − 0.281i)8-s + (−0.5 − 0.866i)9-s + (0.981 − 0.189i)10-s + (0.981 + 0.189i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (−0.786 + 0.618i)16-s + (0.0475 + 0.998i)17-s + (−0.995 − 0.0950i)18-s + (0.0475 − 0.998i)19-s + (0.415 − 0.909i)20-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.5 + 0.866i)3-s + (−0.327 − 0.945i)4-s + (0.723 + 0.690i)5-s + (0.415 + 0.909i)6-s + (−0.959 − 0.281i)8-s + (−0.5 − 0.866i)9-s + (0.981 − 0.189i)10-s + (0.981 + 0.189i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (−0.786 + 0.618i)16-s + (0.0475 + 0.998i)17-s + (−0.995 − 0.0950i)18-s + (0.0475 − 0.998i)19-s + (0.415 − 0.909i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034284616 + 0.7353160353i\)
\(L(\frac12)\) \(\approx\) \(1.034284616 + 0.7353160353i\)
\(L(1)\) \(\approx\) \(1.121050775 + 0.05196274525i\)
\(L(1)\) \(\approx\) \(1.121050775 + 0.05196274525i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.580 - 0.814i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.723 + 0.690i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
23 \( 1 + (-0.786 + 0.618i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.327 + 0.945i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.995 + 0.0950i)T \)
53 \( 1 + (-0.786 - 0.618i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.928 + 0.371i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−22.31587403554979235543031653657, −21.2797325007864893676206966850, −20.55150435868124181723685025918, −19.570773693449549592505749642554, −18.32961799593847074120726003463, −17.817620286288169250105670341988, −17.11972642469727302309401659383, −16.42957800873101427785932600495, −15.71505261787641502100714048602, −14.36106822603220545643839329428, −13.900660242155900696119809556991, −13.081962536479331759918318536044, −12.28131324909177570615304073350, −11.94758442040147440239164865625, −10.44971927085070944941991578833, −9.458668194659431917691591850584, −8.27912731132225013683103052887, −7.79065933244723984906963081771, −6.68721002237091399093770582015, −5.985498348738117191744261751359, −5.227464651427070748792458316461, −4.54303242760888784389976695773, −2.9799315822496870668550209721, −1.99381485989919997537401046054, −0.49096618838612021442682461821, 1.44678443794241984416315290854, 2.59071374322298473713418951653, 3.42222909347870516805902459677, 4.4727428458803034867306069699, 5.16552711122391575596129442103, 6.19872895660787109129380001801, 6.73073630630794849484448705985, 8.53792585391659521226003934141, 9.701929510016165320500161302123, 9.92718988152041309904810246466, 10.88546365710306498392761600748, 11.528841207179364834007037385028, 12.330305980713929887630760820256, 13.44605322394153290732310809971, 14.16577375830324722442385806881, 14.9408234889385098817689375150, 15.53929660919305611560646265928, 16.75275025215991712545903614514, 17.60073507677704250928580138040, 18.206886801292755200345558730166, 19.35314635478800846938786698270, 19.90692620309212209302507519799, 21.1619369714350103624402577225, 21.46179643817375450797863108833, 22.098412514166626615268474898326

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