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Properties

Label 1-2e7-128.37-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: $\chi_{128} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8197934018 - 0.3162292106i\)
\(L(\frac12)\) \(\approx\) \(0.8197934018 - 0.3162292106i\)
\(L(1)\) \(\approx\) \(0.8979925624 - 0.1104947375i\)
\(L(1)\) \(\approx\) \(0.8979925624 - 0.1104947375i\)

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

−29.005769631716109730306668039491, −28.16285070104466298826274046776, −26.85993641163801572706453343855, −25.840215099578072791392949966924, −24.81236210088639125142752795155, −23.89068587699783905714270125600, −22.95634254041507138446577829650, −21.88716018878250599700433483441, −21.15965459243790178641894571794, −19.20298786609522424535145101458, −18.7272140460613967690546162264, −17.91682237488464332477965849705, −16.78167967863743794992986848227, −15.44875732428398322941096757442, −14.27995475011521243272645640329, −13.32854613338059457720839324730, −11.9482233037046092977935921205, −11.286262255258300848792658258217, −10.04046078704206696768442694851, −8.38510655906230258818021177688, −7.29523178408705971837485664406, −6.13190045919587093410665236177, −5.224831426296574367421062550, −3.02348605305100007719004690663, −1.81877992237188662803580961207, 0.933733739251696903321768474103, 3.27882005118281379283468230253, 4.92500071733919806141769576997, 5.20002229761003212326479612598, 7.1411381995106049694875026308, 8.462434201105493830457470912470, 9.84151956023972734427551830577, 10.51110063839163059317347341298, 11.87550714637299707618098277443, 12.92337736915650528467201432608, 14.18302962169019295730656795310, 15.53123858882481279524047365801, 16.35892669529156780678224208700, 17.31810442348505519678567380897, 18.05573263940233305375581204625, 20.07360399189397235531081874065, 20.57938373772452732955257513874, 21.41146501931115510556402491580, 22.8348038586319647418719568971, 23.428388042089256556361467934913, 24.60927642195549319599876286183, 25.76439884477925358471589540582, 27.01027129442356407011741013487, 27.55181375449127682169110511243, 28.71570544566501761380112439075

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