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Properties

Label 2-67-67.25-c1-0-2
Degree $2$
Conductor $67$
Sign $0.241 - 0.970i$
Analytic cond. $0.534997$
Root an. cond. $0.731435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.357 + 2.48i)2-s + (1.75 − 2.02i)3-s + (−4.14 + 1.21i)4-s + (−1.41 + 0.909i)5-s + (5.67 + 3.64i)6-s + (−0.198 − 1.38i)7-s + (−2.41 − 5.29i)8-s + (−0.597 − 4.15i)9-s + (−2.76 − 3.19i)10-s + (−3.98 + 2.56i)11-s + (−4.81 + 10.5i)12-s + (2.48 − 5.43i)13-s + (3.37 − 0.989i)14-s + (−0.642 + 4.46i)15-s + (5.04 − 3.24i)16-s + (4.13 + 1.21i)17-s + ⋯
L(s)  = 1  + (0.252 + 1.75i)2-s + (1.01 − 1.17i)3-s + (−2.07 + 0.608i)4-s + (−0.632 + 0.406i)5-s + (2.31 + 1.48i)6-s + (−0.0751 − 0.522i)7-s + (−0.855 − 1.87i)8-s + (−0.199 − 1.38i)9-s + (−0.875 − 1.01i)10-s + (−1.20 + 0.772i)11-s + (−1.38 + 3.04i)12-s + (0.688 − 1.50i)13-s + (0.900 − 0.264i)14-s + (−0.165 + 1.15i)15-s + (1.26 − 0.811i)16-s + (1.00 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(67\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(0.534997\)
Root analytic conductor: \(0.731435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 67,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.854798 + 0.668397i\)
\(L(\frac12)\) \(\approx\) \(0.854798 + 0.668397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 + (-8.16 + 0.533i)T \)
good2 \( 1 + (-0.357 - 2.48i)T + (-1.91 + 0.563i)T^{2} \)
3 \( 1 + (-1.75 + 2.02i)T + (-0.426 - 2.96i)T^{2} \)
5 \( 1 + (1.41 - 0.909i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.198 + 1.38i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (3.98 - 2.56i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-2.48 + 5.43i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-4.13 - 1.21i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (0.661 - 4.59i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (0.704 - 0.812i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 - 2.39T + 29T^{2} \)
31 \( 1 + (1.12 + 2.45i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + 3.75T + 37T^{2} \)
41 \( 1 + (0.434 + 0.127i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (4.00 + 1.17i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (0.0399 - 0.0461i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (-3.76 + 1.10i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-0.612 - 1.34i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (3.84 + 2.47i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (-0.761 + 0.223i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-2.64 - 1.70i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-0.0424 + 0.0929i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (11.6 - 7.45i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (10.0 + 11.5i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.07560078400965846428182138375, −14.15561478635835442046175376875, −13.22644883752387693568656090053, −12.55727134095365936172516770282, −10.21625696762879793691884765264, −8.278050281860148019306705049567, −7.82181458087893774970367942312, −7.12128951269144257203909334860, −5.62664234107116793846572829404, −3.51858143632990563734834586836, 2.71849616361170820778534945103, 3.86375201345990309193824127170, 4.97862567815134185515754925577, 8.427568450465310347677889528053, 9.080156877192285211255296116344, 10.17870856353478654327032975025, 11.18169395939950859676233422977, 12.16832345078095323079736707227, 13.49838604035617498329026562991, 14.21336334402321268409331626036

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