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Properties

Label 1-667-667.99-r0-0-0
Degree $1$
Conductor $667$
Sign $0.394 + 0.918i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.989 − 0.142i)2-s + (0.909 + 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.654 + 0.755i)5-s + (0.959 + 0.281i)6-s + (−0.841 + 0.540i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (−0.540 + 0.841i)10-s + (0.989 + 0.142i)11-s + (0.989 + 0.142i)12-s + (−0.841 − 0.540i)13-s + (−0.755 + 0.654i)14-s + (−0.909 + 0.415i)15-s + (0.841 − 0.540i)16-s + (−0.281 + 0.959i)17-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)2-s + (0.909 + 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.654 + 0.755i)5-s + (0.959 + 0.281i)6-s + (−0.841 + 0.540i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (−0.540 + 0.841i)10-s + (0.989 + 0.142i)11-s + (0.989 + 0.142i)12-s + (−0.841 − 0.540i)13-s + (−0.755 + 0.654i)14-s + (−0.909 + 0.415i)15-s + (0.841 − 0.540i)16-s + (−0.281 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.394 + 0.918i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.394 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.582858494 + 1.701175339i\)
\(L(\frac12)\) \(\approx\) \(2.582858494 + 1.701175339i\)
\(L(1)\) \(\approx\) \(2.093562999 + 0.6324798804i\)
\(L(1)\) \(\approx\) \(2.093562999 + 0.6324798804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.989 - 0.142i)T \)
3 \( 1 + (0.909 + 0.415i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (0.989 + 0.142i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
31 \( 1 + (-0.909 + 0.415i)T \)
37 \( 1 + (0.755 - 0.654i)T \)
41 \( 1 + (0.755 + 0.654i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.909 - 0.415i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.540 - 0.841i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.909 - 0.415i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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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

−22.584108734400227364587714201095, −21.99606817810583055719575463666, −20.81932522759656728123805925314, −20.137222406801811527377295735934, −19.64591953333853017649803441632, −19.07335597695118460087959600507, −17.48789993101379415021758784296, −16.51082990473264326529649011349, −15.97356931577278225166437490733, −15.003658759887430113138077950086, −14.24255208799599631766207889358, −13.42151262541055885262774368019, −12.83153521974712672340329048428, −11.98473840720220661615916801301, −11.28626457044197206258302092133, −9.64199842459177660688104514833, −9.06770339401621305769996585464, −7.79462268891562505312310060413, −7.10194997782969543673924162141, −6.45901995477916875304475907402, −4.92310727172915699333760237426, −4.12125660906232896920614134605, −3.37006666830631960550989101318, −2.36042480828309848410181256085, −0.999481425458295441872704885768, 1.87525164758519511961123648125, 2.83354034282875489448751039492, 3.60017878793377497514887405186, 4.175495957153603515423287986086, 5.51770283310493306801860834291, 6.569744022008459508780289110603, 7.35752412612191185178686639781, 8.33245823518658840198240722336, 9.61676435253961053358485230653, 10.270311809160597318191342772177, 11.25815628904579957769297792627, 12.31783647280773648789771516327, 12.837838770464565641716560026020, 14.03518845968615471856333814617, 14.78839380966468290278725656674, 15.088258848179797294378495136433, 16.00990849933312913006475378188, 16.75638372279384756865267766819, 18.37373116637788963546301765701, 19.40393645729534594427776665182, 19.634953057377523327799352811008, 20.376065791366334549380038615896, 21.703365925039359565041709709018, 21.998311946032450797775175886774, 22.70166074592660144572304011289

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