↑

Properties

Label 1-344-344.205-r1-0-0
Degree $1$
Conductor $344$
Sign $0.954 - 0.298i$
Analytic cond. $36.9679$
Root an. cond. $36.9679$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0747 + 0.997i)3-s + (0.955 − 0.294i)5-s + (0.5 − 0.866i)7-s + (−0.988 + 0.149i)9-s + (−0.623 − 0.781i)11-s + (0.733 + 0.680i)13-s + (0.365 + 0.930i)15-s + (0.955 + 0.294i)17-s + (−0.988 − 0.149i)19-s + (0.900 + 0.433i)21-s + (0.365 − 0.930i)23-s + (0.826 − 0.563i)25-s + (−0.222 − 0.974i)27-s + (0.0747 − 0.997i)29-s + (0.826 + 0.563i)31-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)3-s + (0.955 − 0.294i)5-s + (0.5 − 0.866i)7-s + (−0.988 + 0.149i)9-s + (−0.623 − 0.781i)11-s + (0.733 + 0.680i)13-s + (0.365 + 0.930i)15-s + (0.955 + 0.294i)17-s + (−0.988 − 0.149i)19-s + (0.900 + 0.433i)21-s + (0.365 − 0.930i)23-s + (0.826 − 0.563i)25-s + (−0.222 − 0.974i)27-s + (0.0747 − 0.997i)29-s + (0.826 + 0.563i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.954 - 0.298i$
Analytic conductor: \(36.9679\)
Root analytic conductor: \(36.9679\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (1:\ ),\ 0.954 - 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.413146451 - 0.3685260010i\)
\(L(\frac12)\) \(\approx\) \(2.413146451 - 0.3685260010i\)
\(L(1)\) \(\approx\) \(1.367746139 + 0.07265180198i\)
\(L(1)\) \(\approx\) \(1.367746139 + 0.07265180198i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.0747 + 0.997i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (-0.988 - 0.149i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (0.826 + 0.563i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (0.988 + 0.149i)T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.0747 - 0.997i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.991051671729529414491143048461, −23.81000274040688550542843637089, −23.09832731556078783671615631159, −22.115422381054346673279045278452, −21.02418702420727241106740574499, −20.490264244220763485504110468696, −19.0155881222488338517054189245, −18.46781739359795971974275236501, −17.72609889320712130284422956368, −17.07389914642022030477361062286, −15.488599825383649378169863721783, −14.70187848313639232397222853004, −13.74510996434558738247210352324, −12.91256136926170531856099211034, −12.160461358989255430969892308122, −11.019100074978766291603834304938, −9.97997104731390452914625624745, −8.792256378759239175408487229182, −7.94723550450739583765780738446, −6.86210850233213737680254149803, −5.805174782005812662186092074134, −5.175787704460091418324692692347, −3.11029751163829433558901940915, −2.19098412459483893073682995516, −1.250066328915178829771576513643, 0.74575995761542148801466959451, 2.25182643648491445678401563386, 3.612824050792915282973497158705, 4.61271021601745766867941037356, 5.56184301820496617562144419175, 6.56084137725837422875876535774, 8.237563803440320413802298199250, 8.8186581702912091676613399877, 10.165812807211015604918451794214, 10.53102330520861889185013497758, 11.57659316218069888340492872899, 13.08275763459530153574716175769, 13.91943747931895758196018717322, 14.52849245629044958837969093093, 15.772430959851606062986200805057, 16.77258900063433551924140659877, 17.07222877971651021807916975036, 18.317540930975572019524989524792, 19.40229944600627239782383816478, 20.64572815403341426278503055644, 21.14861340855310539699552643497, 21.54893492336393597047464459449, 22.91656241441695208797325577289, 23.61945911203139076343158223275, 24.69408613624184154184073554783

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