Properties

Label 2-68-68.67-c4-0-27
Degree $2$
Conductor $68$
Sign $1$
Analytic cond. $7.02915$
Root an. cond. $2.65125$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·3-s + 16·4-s + 64·6-s − 64·7-s + 64·8-s + 175·9-s − 208·11-s + 256·12-s − 274·13-s − 256·14-s + 256·16-s + 289·17-s + 700·18-s − 1.02e3·21-s − 832·22-s + 608·23-s + 1.02e3·24-s + 625·25-s − 1.09e3·26-s + 1.50e3·27-s − 1.02e3·28-s − 256·31-s + 1.02e3·32-s − 3.32e3·33-s + 1.15e3·34-s + 2.80e3·36-s + ⋯
L(s)  = 1  + 2-s + 16/9·3-s + 4-s + 16/9·6-s − 1.30·7-s + 8-s + 2.16·9-s − 1.71·11-s + 16/9·12-s − 1.62·13-s − 1.30·14-s + 16-s + 17-s + 2.16·18-s − 2.32·21-s − 1.71·22-s + 1.14·23-s + 16/9·24-s + 25-s − 1.62·26-s + 2.06·27-s − 1.30·28-s − 0.266·31-s + 32-s − 3.05·33-s + 34-s + 2.16·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(7.02915\)
Root analytic conductor: \(2.65125\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{68} (67, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.199262322\)
\(L(\frac12)\) \(\approx\) \(4.199262322\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
17 \( 1 - p^{2} T \)
good3 \( 1 - 16 T + p^{4} T^{2} \)
5 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
7 \( 1 + 64 T + p^{4} T^{2} \)
11 \( 1 + 208 T + p^{4} T^{2} \)
13 \( 1 + 274 T + p^{4} T^{2} \)
19 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
23 \( 1 - 608 T + p^{4} T^{2} \)
29 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
31 \( 1 + 256 T + p^{4} T^{2} \)
37 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( 1 + 4174 T + p^{4} T^{2} \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
67 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
71 \( 1 - 7904 T + p^{4} T^{2} \)
73 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
79 \( 1 + 2656 T + p^{4} T^{2} \)
83 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
89 \( 1 - 542 T + p^{4} T^{2} \)
97 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
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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

−14.05010141149289878170972719044, −12.87826701888610965832876178424, −12.68475086515051156452333900299, −10.39282183780405808912745098959, −9.542059665056304683511669526807, −7.88172993268769447535091850721, −7.03925713427998392994883296604, −5.01826149191963137119078195992, −3.21632357021810244262800524064, −2.59797013717096346063582750689, 2.59797013717096346063582750689, 3.21632357021810244262800524064, 5.01826149191963137119078195992, 7.03925713427998392994883296604, 7.88172993268769447535091850721, 9.542059665056304683511669526807, 10.39282183780405808912745098959, 12.68475086515051156452333900299, 12.87826701888610965832876178424, 14.05010141149289878170972719044

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