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Properties

Label 1-2e7-128.109-r0-0-0
Degree $1$
Conductor $128$
Sign $0.740 + 0.671i$
Analytic cond. $0.594429$
Root an. cond. $0.594429$
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  + (0.555 + 0.831i)3-s + (−0.195 − 0.980i)5-s + (0.382 + 0.923i)7-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.555 + 0.831i)21-s + (0.923 + 0.382i)23-s + (−0.923 + 0.382i)25-s + (−0.980 + 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s + ⋯
L(s)  = 1  + (0.555 + 0.831i)3-s + (−0.195 − 0.980i)5-s + (0.382 + 0.923i)7-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.555 + 0.831i)21-s + (0.923 + 0.382i)23-s + (−0.923 + 0.382i)25-s + (−0.980 + 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(0.594429\)
Root analytic conductor: \(0.594429\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ 0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.197806620 + 0.4620449997i\)
\(L(\frac12)\) \(\approx\) \(1.197806620 + 0.4620449997i\)
\(L(1)\) \(\approx\) \(1.202096253 + 0.2852291523i\)
\(L(1)\) \(\approx\) \(1.202096253 + 0.2852291523i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.555 + 0.831i)T \)
5 \( 1 + (-0.195 - 0.980i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (0.831 + 0.555i)T \)
13 \( 1 + (0.195 - 0.980i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.980 - 0.195i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (0.555 - 0.831i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.831 - 0.555i)T \)
59 \( 1 + (0.195 + 0.980i)T \)
61 \( 1 + (-0.555 - 0.831i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.980 + 0.195i)T \)
89 \( 1 + (0.923 - 0.382i)T \)
97 \( 1 - iT \)
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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

−29.05922440786221996891876679390, −27.33400661926823284123905399383, −26.64266688866419460213709286574, −25.710297011289610423663477187198, −24.70443152308035266523402513913, −23.57204803235639706617769056184, −22.98786496348618945933625567014, −21.52478608696412270081183317240, −20.44882078278122416806256879866, −19.23790789345696482239533006078, −18.79383048366331008196484769093, −17.54465631740588006166019898088, −16.50047588667096818904431720972, −14.69413833412592146006111737299, −14.289579708513733609847637129816, −13.28091701730342714300558898452, −11.7794874940000526375893805006, −10.96082409703729294738904412127, −9.451385715284216720990715367353, −8.13440786544419540012568730224, −7.07223439849927364303229973558, −6.34417687794580114112361703055, −4.13577637136009243721556961790, −2.988183336231101290169476202095, −1.40615458747066348951707210442, 1.86571006722425331926189975166, 3.541724851697954470605827248883, 4.74611296960581491549128962541, 5.744167003935538363361460654290, 7.859590566880932829090742498803, 8.75770443809423603389187780515, 9.55569043883925440225346876890, 10.95731794958018390607540675000, 12.232816542875914465083257433162, 13.22091168685038300270266071472, 14.91121689350982518051385341707, 15.18565340227105819944218516526, 16.57239128807697120106584884873, 17.382591245297233596401975991687, 18.98906357044461270237488935893, 19.975086170718308419729053463916, 20.82800356578739309619800715459, 21.64040047253024834078373742592, 22.71714812645858100098839264806, 24.06586459756565608443229507748, 25.23202354749974399795352156966, 25.56896379366931097629364478899, 27.34999250890798027371745206861, 27.73035137398556643613163842398, 28.461463648413178508036272341571

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