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Properties

Label 1-67-67.39-r0-0-0
Degree $1$
Conductor $67$
Sign $0.994 + 0.107i$
Analytic cond. $0.311146$
Root an. cond. $0.311146$
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  + (0.723 − 0.690i)2-s + (−0.142 + 0.989i)3-s + (0.0475 − 0.998i)4-s + (0.415 + 0.909i)5-s + (0.580 + 0.814i)6-s + (0.235 + 0.971i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (0.928 + 0.371i)10-s + (0.580 − 0.814i)11-s + (0.981 + 0.189i)12-s + (−0.327 − 0.945i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.0475 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (−0.142 + 0.989i)3-s + (0.0475 − 0.998i)4-s + (0.415 + 0.909i)5-s + (0.580 + 0.814i)6-s + (0.235 + 0.971i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (0.928 + 0.371i)10-s + (0.580 − 0.814i)11-s + (0.981 + 0.189i)12-s + (−0.327 − 0.945i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.0475 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.994 + 0.107i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.994 + 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.285710053 + 0.06947727385i\)
\(L(\frac12)\) \(\approx\) \(1.285710053 + 0.06947727385i\)
\(L(1)\) \(\approx\) \(1.375658067 + 0.002938546210i\)
\(L(1)\) \(\approx\) \(1.375658067 + 0.002938546210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.723 - 0.690i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (0.235 + 0.971i)T \)
11 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (-0.327 - 0.945i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.928 - 0.371i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (0.0475 - 0.998i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−31.88930111858292451180171429417, −31.06887451430871737057589835080, −29.84611332323244177720652831510, −29.189959861448969549604262307775, −27.60519661585975021004266566455, −26.051851103688336539582390744231, −25.05834338077038012901880372195, −24.22519542111154857006103410394, −23.435220218633079039174739353601, −22.36146580562618024998569969501, −20.78934380468195502400504097745, −19.95187194699357613948516445942, −18.0660900446695629985550148424, −17.06761135433588216867855147154, −16.3960137394440873254397968796, −14.3607638379334839641883848772, −13.717139261910412673472914259364, −12.54553074261932726160697327457, −11.661275253112375854393950111249, −9.33559226027715656337045454270, −7.78668819441890235290321980588, −6.89099922970307322569058551936, −5.47021131668229920789699471945, −4.14327889509968379862388899508, −1.8067559837071901945037770574, 2.50285139473869148785406473973, 3.64055104245760379597111814691, 5.33550454595489970311731070427, 6.21319983321410020918747539369, 8.828369446408544747540899103, 10.14073460213497345369966012665, 11.01641860941125686249951535107, 12.110245298622091246696670053261, 13.825301177860578348610189437825, 14.88003272075852225489714449811, 15.55651830152480555004284668864, 17.43071172667464972124071032639, 18.74005212351601924176457312697, 19.93855120098455713668697556236, 21.30293832842658095669985982854, 22.03269585293754624945037368432, 22.50095820619385464293743327732, 24.12253363410172873141452255439, 25.43569097543767091577451362037, 26.816076258413186182820580562125, 27.844172329187596652780065997498, 28.75793542242878592491420414392, 29.99494865889214047182567753131, 30.88039817380503833967838296152, 32.26665587825237868755734915318

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